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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | cii 24901 | Extend class notation with the unit interval. |
| class II | ||
| Syntax | ccncf 24902 | Extend class notation to include the operation which returns a class of continuous complex functions. |
| class –cn→ | ||
| Definition | df-ii 24903 | Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
| ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1)))) | ||
| Definition | df-cncf 24904* | Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.) |
| ⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)}) | ||
| Theorem | iitopon 24905 | The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| ⊢ II ∈ (TopOn‘(0[,]1)) | ||
| Theorem | iitop 24906 | The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ II ∈ Top | ||
| Theorem | iiuni 24907 | The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.) |
| ⊢ (0[,]1) = ∪ II | ||
| Theorem | dfii2 24908 | Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | ||
| Theorem | dfii3 24909 | Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ II = (𝐽 ↾t (0[,]1)) | ||
| Theorem | dfii4 24910 | Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| ⊢ 𝐼 = (ℂfld ↾s (0[,]1)) ⇒ ⊢ II = (TopOpen‘𝐼) | ||
| Theorem | dfii5 24911 | The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1)))) | ||
| Theorem | iicmp 24912 | The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.) |
| ⊢ II ∈ Comp | ||
| Theorem | iiconn 24913 | The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ II ∈ Conn | ||
| Theorem | cncfval 24914* | The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.) |
| ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)}) | ||
| Theorem | elcncf 24915* | Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.) |
| ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))) | ||
| Theorem | elcncf2 24916* | Version of elcncf 24915 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.) |
| ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) | ||
| Theorem | cncfrss 24917 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
| ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | ||
| Theorem | cncfrss2 24918 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
| ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | ||
| Theorem | cncff 24919 | A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | ||
| Theorem | cncfi 24920* | Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| ⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) | ||
| Theorem | elcncf1di 24921* | Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)) & ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))) ⇒ ⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))) | ||
| Theorem | elcncf1ii 24922* | Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.) |
| ⊢ 𝐹:𝐴⟶𝐵 & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+) & ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)) | ||
| Theorem | rescncf 24923 | A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| ⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵))) | ||
| Theorem | cncfcdm 24924 | Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.) |
| ⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ 𝐹:𝐴⟶𝐶)) | ||
| Theorem | cncfss 24925 | The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.) |
| ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) | ||
| Theorem | climcncf 24926 | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐺:𝑍⟶𝐴) & ⊢ (𝜑 → 𝐺 ⇝ 𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷)) | ||
| Theorem | abscncf 24927 | Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
| ⊢ abs ∈ (ℂ–cn→ℝ) | ||
| Theorem | recncf 24928 | Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
| ⊢ ℜ ∈ (ℂ–cn→ℝ) | ||
| Theorem | imcncf 24929 | Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
| ⊢ ℑ ∈ (ℂ–cn→ℝ) | ||
| Theorem | cjcncf 24930 | Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
| ⊢ ∗ ∈ (ℂ–cn→ℂ) | ||
| Theorem | mulc1cncf 24931* | Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
| Theorem | divccncf 24932* | Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
| Theorem | cncfco 24933 | The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→𝐶)) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶)) | ||
| Theorem | cncfcompt2 24934* | Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵)) & ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸)) & ⊢ (𝜑 → 𝐵 ⊆ 𝐶) & ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸)) | ||
| Theorem | cncfmet 24935 | Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) |
| ⊢ 𝐶 = ((abs ∘ − ) ↾ (𝐴 × 𝐴)) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝐵 × 𝐵)) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (MetOpen‘𝐷) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾)) | ||
| Theorem | cncfcn 24936 | Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝐴) & ⊢ 𝐿 = (𝐽 ↾t 𝐵) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿)) | ||
| Theorem | cncfcn1 24937 | Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) | ||
| Theorem | cncfmptc 24938* | A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.) |
| ⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇)) | ||
| Theorem | cncfmptid 24939* | The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.) |
| ⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇)) | ||
| Theorem | cncfmpt1f 24940* | Composition of continuous functions. –cn→ analogue of cnmpt11f 23672. (Contributed by Mario Carneiro, 3-Sep-2014.) |
| ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ)) | ||
| Theorem | cncfmpt2f 24941* | Composition of continuous functions. –cn→ analogue of cnmpt12f 23674. (Contributed by Mario Carneiro, 3-Sep-2014.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ)) | ||
| Theorem | cncfmpt2ss 24942* | Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆)) & ⊢ 𝑆 ⊆ ℂ & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆)) | ||
| Theorem | addccncf 24943* | Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
| Theorem | idcncf 24944 | The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 24939 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥) ⇒ ⊢ 𝐹 ∈ (ℂ–cn→ℂ) | ||
| Theorem | sub1cncf 24945* | Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
| Theorem | sub2cncf 24946* | Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
| Theorem | cdivcncf 24947* | Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) |
| ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ((ℂ ∖ {0})–cn→ℂ)) | ||
| Theorem | negcncf 24948* | The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ)) | ||
| Theorem | negcncfOLD 24949* | Obsolete version of negcncf 24948 as of 9-Apr-2025. (Contributed by Mario Carneiro, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ)) | ||
| Theorem | negfcncf 24950* | The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
| ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥)) ⇒ ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ)) | ||
| Theorem | abscncfALT 24951 | Absolute value is continuous. Alternate proof of abscncf 24927. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ abs ∈ (ℂ–cn→ℝ) | ||
| Theorem | cncfcnvcn 24952 | Rewrite cmphaushmeo 23808 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑋) ⇒ ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋))) | ||
| Theorem | expcncf 24953* | The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 24896. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) | ||
| Theorem | cnmptre 24954* | Lemma for iirevcn 24957 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.) |
| ⊢ 𝑅 = (TopOpen‘ℂfld) & ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) & ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | cnmpopc 24955* | Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.) |
| ⊢ 𝑅 = (topGen‘ran (,)) & ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵)) & ⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶)) & ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶)) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸) & ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾)) & ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾)) | ||
| Theorem | iirev 24956 | Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1)) | ||
| Theorem | iirevcn 24957 | The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.) |
| ⊢ (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn II) | ||
| Theorem | iihalf1 24958 | Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1)) | ||
| Theorem | iihalf1cn 24959 | The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2))) ⇒ ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II) | ||
| Theorem | iihalf1cnOLD 24960 | Obsolete version of iihalf1cn 24959 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2))) ⇒ ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II) | ||
| Theorem | iihalf2 24961 | Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ (𝑋 ∈ ((1 / 2)[,]1) → ((2 · 𝑋) − 1) ∈ (0[,]1)) | ||
| Theorem | iihalf2cn 24962 | The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) ⇒ ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) | ||
| Theorem | iihalf2cnOLD 24963 | Obsolete version of iihalf2cn 24962 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) ⇒ ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) | ||
| Theorem | elii1 24964 | Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.) |
| ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) | ||
| Theorem | elii2 24965 | Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.) |
| ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1)) | ||
| Theorem | iimulcl 24966 | The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ ((𝐴 ∈ (0[,]1) ∧ 𝐵 ∈ (0[,]1)) → (𝐴 · 𝐵) ∈ (0[,]1)) | ||
| Theorem | iimulcn 24967* | Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II) | ||
| Theorem | iimulcnOLD 24968* | Obsolete version of iimulcn 24967 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II) | ||
| Theorem | icoopnst 24969 | A half-open interval starting at 𝐴 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
| ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽)) | ||
| Theorem | iocopnst 24970 | A half-open interval ending at 𝐵 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) |
| ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽)) | ||
| Theorem | icchmeo 24971* | The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵)))) | ||
| Theorem | icchmeoOLD 24972* | Obsolete version of icchmeo 24971 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ ((𝑥 · 𝐵) + ((1 − 𝑥) · 𝐴))) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐹 ∈ (IIHomeo(𝐽 ↾t (𝐴[,]𝐵)))) | ||
| Theorem | icopnfcnv 24973* | Define a bijection from [0, 1) to [0, +∞). (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) ⇒ ⊢ (𝐹:(0[,)1)–1-1-onto→(0[,)+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦)))) | ||
| Theorem | icopnfhmeo 24974* | The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 ↾t (0[,)1))Homeo(𝐽 ↾t (0[,)+∞)))) | ||
| Theorem | iccpnfcnv 24975* | Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))) ⇒ ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦))))) | ||
| Theorem | iccpnfhmeo 24976 | The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))) & ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ⇒ ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾)) | ||
| Theorem | xrhmeo 24977* | The bijection from [-1, 1] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥)))) & ⊢ 𝐺 = (𝑦 ∈ (-1[,]1) ↦ if(0 ≤ 𝑦, (𝐹‘𝑦), -𝑒(𝐹‘-𝑦))) & ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽 ↾t (-1[,]1))Homeo(ordTop‘ ≤ ))) | ||
| Theorem | xrhmph 24978 | The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ II ≃ (ordTop‘ ≤ ) | ||
| Theorem | xrcmp 24979 | The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 24828), this means that ℝ* is a compactification of ℝ. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (ordTop‘ ≤ ) ∈ Comp | ||
| Theorem | xrconn 24980 | The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (ordTop‘ ≤ ) ∈ Conn | ||
| Theorem | icccvx 24981 | A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ (𝐴[,]𝐵) ∧ 𝐷 ∈ (𝐴[,]𝐵) ∧ 𝑇 ∈ (0[,]1)) → (((1 − 𝑇) · 𝐶) + (𝑇 · 𝐷)) ∈ (𝐴[,]𝐵))) | ||
| Theorem | oprpiece1res1 24982* | Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 ≤ 𝐵 & ⊢ 𝑅 ∈ V & ⊢ 𝑆 ∈ V & ⊢ 𝐾 ∈ (𝐴[,]𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) & ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐶 ↦ 𝑅) ⇒ ⊢ (𝐹 ↾ ((𝐴[,]𝐾) × 𝐶)) = 𝐺 | ||
| Theorem | oprpiece1res2 24983* | Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.) |
| ⊢ 𝐴 ∈ ℝ & ⊢ 𝐵 ∈ ℝ & ⊢ 𝐴 ≤ 𝐵 & ⊢ 𝑅 ∈ V & ⊢ 𝑆 ∈ V & ⊢ 𝐾 ∈ (𝐴[,]𝐵) & ⊢ 𝐹 = (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝐶 ↦ if(𝑥 ≤ 𝐾, 𝑅, 𝑆)) & ⊢ (𝑥 = 𝐾 → 𝑅 = 𝑃) & ⊢ (𝑥 = 𝐾 → 𝑆 = 𝑄) & ⊢ (𝑦 ∈ 𝐶 → 𝑃 = 𝑄) & ⊢ 𝐺 = (𝑥 ∈ (𝐾[,]𝐵), 𝑦 ∈ 𝐶 ↦ 𝑆) ⇒ ⊢ (𝐹 ↾ ((𝐾[,]𝐵) × 𝐶)) = 𝐺 | ||
| Theorem | cnrehmeo 24984* | The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 13027 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ × ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) | ||
| Theorem | cnrehmeoOLD 24985* | Obsolete version of cnrehmeo 24984 as of 9-Apr-2025. (Contributed by Mario Carneiro, 25-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) | ||
| Theorem | cnheiborlem 24986* | Lemma for cnheibor 24987. (Contributed by Mario Carneiro, 14-Sep-2014.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑇 = (𝐽 ↾t 𝑋) & ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) & ⊢ 𝑌 = (𝐹 “ ((-𝑅[,]𝑅) × (-𝑅[,]𝑅))) ⇒ ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑅 ∈ ℝ ∧ ∀𝑧 ∈ 𝑋 (abs‘𝑧) ≤ 𝑅)) → 𝑇 ∈ Comp) | ||
| Theorem | cnheibor 24987* | Heine-Borel theorem for complex numbers. A subset of ℂ is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑇 = (𝐽 ↾t 𝑋) ⇒ ⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))) | ||
| Theorem | cnllycmp 24988 | The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ 𝑛-Locally Comp | ||
| Theorem | rellycmp 24989 | The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ (topGen‘ran (,)) ∈ 𝑛-Locally Comp | ||
| Theorem | bndth 24990* | The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -𝐹.) (Contributed by Mario Carneiro, 12-Aug-2014.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥) | ||
| Theorem | evth 24991* | The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) | ||
| Theorem | evth2 24992* | The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) | ||
| Theorem | lebnumlem1 24993* | Lemma for lebnum 24996. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ⇒ ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+) | ||
| Theorem | lebnumlem2 24994* | Lemma for lebnum 24996. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24878, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) & ⊢ 𝐾 = (topGen‘ran (,)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | lebnumlem3 24995* | Lemma for lebnum 24996. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / ♯(𝑈), since for each 𝑢 ∈ 𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑 ≤ 𝑑(𝑥, 𝑋 ∖ 𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢 ∈ 𝑈𝑑(𝑥, 𝑋 ∖ 𝑢) = 𝐹(𝑥) < Σ𝑢 ∈ 𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) & ⊢ (𝜑 → 𝑈 ∈ Fin) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) & ⊢ 𝐾 = (topGen‘ran (,)) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢) | ||
| Theorem | lebnum 24996* | The Lebesgue number lemma, or Lebesgue covering lemma. If 𝑋 is a compact metric space and 𝑈 is an open cover of 𝑋, then there exists a positive real number 𝑑 such that every ball of size 𝑑 (and every subset of a ball of size 𝑑, including every subset of diameter less than 𝑑) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢) | ||
| Theorem | xlebnum 24997* | Generalize lebnum 24996 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → 𝑋 = ∪ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢) | ||
| Theorem | lebnumii 24998* | Specialize the Lebesgue number lemma lebnum 24996 to the closed unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢) | ||
| Syntax | chtpy 24999 | Extend class notation with the class of homotopies between two continuous functions. |
| class Htpy | ||
| Syntax | cphtpy 25000 | Extend class notation with the class of path homotopies between two continuous functions. |
| class PHtpy | ||
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