![]() |
Metamath
Proof Explorer Theorem List (p. 245 of 479) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30171) |
![]() (30172-31694) |
![]() (31695-47852) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfii5 24401 | The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1)))) | ||
Theorem | iicmp 24402 | The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.) |
⊢ II ∈ Comp | ||
Theorem | iiconn 24403 | The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ II ∈ Conn | ||
Theorem | cncfval 24404* | 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 24405* | 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 24406* | Version of elcncf 24405 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 24407 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | ||
Theorem | cncfrss2 24408 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | ||
Theorem | cncff 24409 | 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 24410* | 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 24411* | 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 24412* | 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 24413 | 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 24414 | 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 24415 | 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 24416 | 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 24417 | Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
⊢ abs ∈ (ℂ–cn→ℝ) | ||
Theorem | recncf 24418 | Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
⊢ ℜ ∈ (ℂ–cn→ℝ) | ||
Theorem | imcncf 24419 | Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
⊢ ℑ ∈ (ℂ–cn→ℝ) | ||
Theorem | cjcncf 24420 | Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.) |
⊢ ∗ ∈ (ℂ–cn→ℂ) | ||
Theorem | mulc1cncf 24421* | 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 24422* | 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 24423 | 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 24424* | 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 24425 | 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 24426 | 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 24427 | 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 24428* | 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 24429* | 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 24430* | Composition of continuous functions. –cn→ analogue of cnmpt11f 23168. (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 24431* | Composition of continuous functions. –cn→ analogue of cnmpt12f 23170. (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 24432* | 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 24433* | 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 24434 | The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 24429 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 24435* | 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 24436* | 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 24437* | 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 24438* | The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥) ⇒ ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ)) | ||
Theorem | negfcncf 24439* | 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 24440 | Absolute value is continuous. Alternate proof of abscncf 24417. (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 24441 | Rewrite cmphaushmeo 23304 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 24442* | The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 24388. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) | ||
Theorem | cnmptre 24443* | Lemma for iirevcn 24446 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 24444* | 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 24445 | Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1)) | ||
Theorem | iirevcn 24446 | 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 24447 | 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 24448 | The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) |
⊢ 𝐽 = ((topGen‘ran (,)) ↾t (0[,](1 / 2))) ⇒ ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (𝐽 Cn II) | ||
Theorem | iihalf2 24449 | 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 24450 | The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) |
⊢ 𝐽 = ((topGen‘ran (,)) ↾t ((1 / 2)[,]1)) ⇒ ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (𝐽 Cn II) | ||
Theorem | elii1 24451 | 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 24452 | 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 24453 | 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 24454* | Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) |
⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥 · 𝑦)) ∈ ((II ×t II) Cn II) | ||
Theorem | icoopnst 24455 | 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 24456 | 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 24457* | The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐵] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) |
⊢ 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 24458* | 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 24459* | 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 24460* | 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 24461 | 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 24462* | 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 24463 | The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ II ≃ (ordTop‘ ≤ ) | ||
Theorem | xrcmp 24464 | The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 24322), this means that ℝ* is a compactification of ℝ. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ (ordTop‘ ≤ ) ∈ Comp | ||
Theorem | xrconn 24465 | The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ (ordTop‘ ≤ ) ∈ Conn | ||
Theorem | icccvx 24466 | 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 24467* | 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 24468* | 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 24469* | The canonical bijection from (ℝ × ℝ) to ℂ described in cnref1o 12969 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.) |
⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦))) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽)Homeo𝐾) | ||
Theorem | cnheiborlem 24470* | Lemma for cnheibor 24471. (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 24471* | 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 24472 | 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 24473 | The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ 𝑛-Locally Comp | ||
Theorem | bndth 24474* | 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 24475* | 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 24476* | 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 24477* | Lemma for lebnum 24480. 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 24478* | Lemma for lebnum 24480. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24372, 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 24479* | Lemma for lebnum 24480. 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 24480* | 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 24481* | Generalize lebnum 24480 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 24482* | Specialize the Lebesgue number lemma lebnum 24480 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 24483 | Extend class notation with the class of homotopies between two continuous functions. |
class Htpy | ||
Syntax | cphtpy 24484 | Extend class notation with the class of path homotopies between two continuous functions. |
class PHtpy | ||
Syntax | cphtpc 24485 | Extend class notation with the path homotopy relation. |
class ≃ph | ||
Definition | df-htpy 24486* | Define the function which takes topological spaces 𝑋, 𝑌 and two continuous functions 𝐹, 𝐺:𝑋⟶𝑌 and returns the class of homotopies from 𝐹 to 𝐺. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ {ℎ ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 ∈ ∪ 𝑥((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))})) | ||
Definition | df-phtpy 24487* | Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 24486) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ {ℎ ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))})) | ||
Theorem | ishtpy 24488* | Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))) | ||
Theorem | htpycn 24489 | A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾)) | ||
Theorem | htpyi 24490 | A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐻0) = (𝐹‘𝐴) ∧ (𝐴𝐻1) = (𝐺‘𝐴))) | ||
Theorem | ishtpyd 24491* | Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾)) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠)) & ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠)) ⇒ ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) | ||
Theorem | htpycom 24492* | Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) & ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) ⇒ ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹)) | ||
Theorem | htpyid 24493* | A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑥)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹)) | ||
Theorem | htpyco1 24494* | Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿)) & ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿)) & ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺)) ⇒ ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃))) | ||
Theorem | htpyco2 24495 | Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.) |
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑃 ∈ (𝐾 Cn 𝐿)) & ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) ⇒ ⊢ (𝜑 → (𝑃 ∘ 𝐻) ∈ ((𝑃 ∘ 𝐹)(𝐽 Htpy 𝐿)(𝑃 ∘ 𝐺))) | ||
Theorem | htpycc 24496* | Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.) |
⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1)))) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐻 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) & ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻)) ⇒ ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻)) | ||
Theorem | isphtpy 24497* | Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))) | ||
Theorem | phtpyhtpy 24498 | A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺)) | ||
Theorem | phtpycn 24499 | A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ ((II ×t II) Cn 𝐽)) | ||
Theorem | phtpyi 24500 | Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.) |
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |