HomeTheorem List (p. 353 of 491)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 353 of 491) | < 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: | Metamath Proof Explorer
(1-30946) |
Hilbert Space Explorer
(30947-32469) |
Users' Mathboxes
(32470-49035) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Syntax | cretr 35201 | Extend class notation with the retract relation. |
| class Retr | ||
| Definition | df-retr 35202* | Define the set of retractions on two topological spaces. We say that 𝑅 is a retraction from 𝐽 to 𝐾. or 𝑅 ∈ (𝐽 Retr 𝐾) iff there is an 𝑆 such that 𝑅:𝐽⟶𝐾, 𝑆:𝐾⟶𝐽 are continuous functions called the retraction and section respectively, and their composite 𝑅 ∘ 𝑆 is homotopic to the identity map. If a retraction exists, we say 𝐽 is a retract of 𝐾. (This terminology is borrowed from HoTT and appears to be nonstandard, although it has similaries to the concept of retract in the category of topological spaces and to a deformation retract in general topology.) Two topological spaces that are retracts of each other are called homotopy equivalent. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪ 𝑗)) ≠ ∅}) | ||
| Syntax | cpconn 35203 | Extend class notation with the class of path-connected topologies. |
| class PConn | ||
| Syntax | csconn 35204 | Extend class notation with the class of simply connected topologies. |
| class SConn | ||
| Definition | df-pconn 35205* | Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the closed unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ PConn = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} | ||
| Definition | df-sconn 35206* | Define the class of simply connected topologies. A topology is simply connected if it is path-connected and every loop (continuous path with identical start and endpoint) is contractible to a point (path-homotopic to a constant function). (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ SConn = {𝑗 ∈ PConn ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} | ||
| Theorem | ispconn 35207* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) | ||
| Theorem | pconncn 35208* | The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵)) | ||
| Theorem | pconntop 35209 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | ||
| Theorem | issconn 35210* | The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | ||
| Theorem | sconnpconn 35211 | A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | ||
| Theorem | sconntop 35212 | A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) | ||
| Theorem | sconnpht 35213 | A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) | ||
| Theorem | cnpconn 35214 | An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ 𝑌 = ∪ 𝐾 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PConn) | ||
| Theorem | pconnconn 35215 | A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Conn) | ||
| Theorem | txpconn 35216 | The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn) | ||
| Theorem | ptpconn 35217 | The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶PConn) → (∏t‘𝐹) ∈ PConn) | ||
| Theorem | indispconn 35218 | The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) |
| ⊢ {∅, 𝐴} ∈ PConn | ||
| Theorem | connpconn 35219 | A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) |
| ⊢ ((𝐽 ∈ Conn ∧ 𝐽 ∈ 𝑛-Locally PConn) → 𝐽 ∈ PConn) | ||
| Theorem | qtoppconn 35220 | A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ PConn) | ||
| Theorem | pconnpi1 35221 | All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝑃 = (𝐽 π1 𝐴) & ⊢ 𝑄 = (𝐽 π1 𝐵) & ⊢ 𝑆 = (Base‘𝑃) & ⊢ 𝑇 = (Base‘𝑄) ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑃 ≃𝑔 𝑄) | ||
| Theorem | sconnpht2 35222 | Any two paths in a simply connected space with the same start and end point are path-homotopic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ (𝜑 → 𝐽 ∈ SConn) & ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0)) & ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1)) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺) | ||
| Theorem | sconnpi1 35223 | A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ PConn ∧ 𝑌 ∈ 𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1o)) | ||
| Theorem | txsconnlem 35224 | Lemma for txsconn 35225. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ (𝜑 → 𝑅 ∈ Top) & ⊢ (𝜑 → 𝑆 ∈ Top) & ⊢ (𝜑 → 𝐹 ∈ (II Cn (𝑅 ×t 𝑆))) & ⊢ 𝐴 = ((1st ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ 𝐵 = ((2nd ↾ (∪ 𝑅 × ∪ 𝑆)) ∘ 𝐹) & ⊢ (𝜑 → 𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)}))) & ⊢ (𝜑 → 𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)}))) ⇒ ⊢ (𝜑 → 𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)})) | ||
| Theorem | txsconn 35225 | The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ ((𝑅 ∈ SConn ∧ 𝑆 ∈ SConn) → (𝑅 ×t 𝑆) ∈ SConn) | ||
| Theorem | cvxpconn 35226* | A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11232. (Revised by GG, 19-Apr-2025.) |
| ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ PConn) | ||
| Theorem | cvxsconn 35227* | A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11232. (Revised by GG, 19-Apr-2025.) |
| ⊢ (𝜑 → 𝑆 ⊆ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆) & ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ (𝜑 → 𝐾 ∈ SConn) | ||
| Theorem | blsconn 35228 | An open ball in the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅) & ⊢ 𝐾 = (𝐽 ↾t 𝑆) ⇒ ⊢ ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SConn) | ||
| Theorem | cnllysconn 35229 | The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Locally SConn | ||
| Theorem | resconn 35230 | A subset of ℝ is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴) ⇒ ⊢ (𝐴 ⊆ ℝ → (𝐽 ∈ SConn ↔ 𝐽 ∈ Conn)) | ||
| Theorem | ioosconn 35231 | An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ ((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn | ||
| Theorem | iccsconn 35232 | A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SConn) | ||
| Theorem | retopsconn 35233 | The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ (topGen‘ran (,)) ∈ SConn | ||
| Theorem | iccllysconn 35234 | A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Locally SConn) | ||
| Theorem | rellysconn 35235 | The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ (topGen‘ran (,)) ∈ Locally SConn | ||
| Theorem | iisconn 35236 | The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ II ∈ SConn | ||
| Theorem | iillysconn 35237 | The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ II ∈ Locally SConn | ||
| Theorem | iinllyconn 35238 | The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.) |
| ⊢ II ∈ 𝑛-Locally Conn | ||
| Syntax | ccvm 35239 | Extend class notation with the class of covering maps. |
| class CovMap | ||
| Definition | df-cvm 35240* | Define the class of covering maps on two topological spaces. A function 𝑓:𝑐⟶𝑗 is a covering map if it is continuous and for every point 𝑥 in the target space there is a neighborhood 𝑘 of 𝑥 and a decomposition 𝑠 of the preimage of 𝑘 as a disjoint union such that 𝑓 is a homeomorphism of each set 𝑢 ∈ 𝑠 onto 𝑘. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))}) | ||
| Theorem | fncvm 35241 | Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ CovMap Fn (Top × Top) | ||
| Theorem | cvmscbv 35242* | Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ 𝑆 = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑐 ∈ 𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑎))))}) | ||
| Theorem | iscvm 35243* | The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) | ||
| Theorem | cvmtop1 35244 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | ||
| Theorem | cvmtop2 35245 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) | ||
| Theorem | cvmcn 35246 | A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽)) | ||
| Theorem | cvmcov 35247* | Property of a covering map. In order to make the covering property more manageable, we define here the set 𝑆(𝑘) of all even coverings of an open set 𝑘 in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) | ||
| Theorem | cvmsrcl 35248* | Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑈 ∈ 𝐽) | ||
| Theorem | cvmsi 35249* | One direction of cvmsval 35250. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))) | ||
| Theorem | cvmsval 35250* | Elementhood in the set 𝑆 of all even coverings of an open set in 𝐽. 𝑆 is an even covering of 𝑈 if it is a nonempty collection of disjoint open sets in 𝐶 whose union is the preimage of 𝑈, such that each set 𝑢 ∈ 𝑆 is homeomorphic under 𝐹 to 𝑈. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝑇 ∈ (𝑆‘𝑈) ↔ (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))) | ||
| Theorem | cvmsss 35251* | An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶) | ||
| Theorem | cvmsn0 35252* | An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ≠ ∅) | ||
| Theorem | cvmsuni 35253* | An even covering of 𝑈 has union equal to the preimage of 𝑈 by 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈)) | ||
| Theorem | cvmsdisj 35254* | An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅)) | ||
| Theorem | cvmshmeo 35255* | Every element of an even covering of 𝑈 is homeomorphic to 𝑈 via 𝐹. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴) ∈ ((𝐶 ↾t 𝐴)Homeo(𝐽 ↾t 𝑈))) | ||
| Theorem | cvmsf1o 35256* | 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝑈) | ||
| Theorem | cvmscld 35257* | The sets of an even covering are clopen in the subspace topology on 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈)))) | ||
| Theorem | cvmsss2 35258* | An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉 ∈ 𝐽 ∧ 𝑉 ⊆ 𝑈) → ((𝑆‘𝑈) ≠ ∅ → (𝑆‘𝑉) ≠ ∅)) | ||
| Theorem | cvmcov2 35259* | The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈 ∈ 𝐽 ∧ 𝑃 ∈ 𝑈) → ∃𝑥 ∈ 𝒫 𝑈(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) | ||
| Theorem | cvmseu 35260* | Every element in ∪ 𝑇 is a member of a unique element of 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → ∃!𝑥 ∈ 𝑇 𝐴 ∈ 𝑥) | ||
| Theorem | cvmsiota 35261* | Identify the unique element of 𝑇 containing 𝐴. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑊 = (℩𝑥 ∈ 𝑇 𝐴 ∈ 𝑥) ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝐵 ∧ (𝐹‘𝐴) ∈ 𝑈)) → (𝑊 ∈ 𝑇 ∧ 𝐴 ∈ 𝑊)) | ||
| Theorem | cvmopnlem 35262* | Lemma for cvmopn 35264. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 ⇒ ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → (𝐹 “ 𝐴) ∈ 𝐽) | ||
| Theorem | cvmfolem 35263* | Lemma for cvmfo 35284. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋) | ||
| Theorem | cvmopn 35264 | A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴 ∈ 𝐶) → (𝐹 “ 𝐴) ∈ 𝐽) | ||
| Theorem | cvmliftmolem1 35265* | Lemma for cvmliftmo 35268. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐾 ∈ Conn) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) & ⊢ (𝜑 → 𝑂 ∈ 𝑌) & ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) & ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) & ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ∈ (𝑆‘𝑈)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝑇) & ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (◡𝑀 “ 𝑊)) & ⊢ ((𝜑 ∧ 𝜓) → (𝐾 ↾t 𝐼) ∈ Conn) & ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐼) & ⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝐼) & ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝐼) & ⊢ ((𝜑 ∧ 𝜓) → (𝐹‘(𝑀‘𝑋)) ∈ 𝑈) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) → 𝑅 ∈ dom (𝑀 ∩ 𝑁))) | ||
| Theorem | cvmliftmolem2 35266* | Lemma for cvmliftmo 35268. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐾 ∈ Conn) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) & ⊢ (𝜑 → 𝑂 ∈ 𝑌) & ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) & ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) & ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) ⇒ ⊢ (𝜑 → 𝑀 = 𝑁) | ||
| Theorem | cvmliftmoi 35267 | A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐾 ∈ Conn) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) & ⊢ (𝜑 → 𝑂 ∈ 𝑌) & ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) & ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) & ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) ⇒ ⊢ (𝜑 → 𝑀 = 𝑁) | ||
| Theorem | cvmliftmo 35268* | A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐾 ∈ Conn) & ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) & ⊢ (𝜑 → 𝑂 ∈ 𝑌) & ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂)) ⇒ ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃)) | ||
| Theorem | cvmliftlem1 35269* | Lemma for cvmlift 35283. In cvmliftlem15 35282, we picked an 𝑁 large enough so that the sections (𝐺 “ [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁]) are all contained in an even covering, and the function 𝑇 enumerates these even coverings. So 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2nd ‘(𝑇‘𝑀) is an even covering of 1st ‘(𝑇‘𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1st ‘(𝑇‘𝑀). (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (2nd ‘(𝑇‘𝑀)) ∈ (𝑆‘(1st ‘(𝑇‘𝑀)))) | ||
| Theorem | cvmliftlem2 35270* | Lemma for cvmlift 35283. 𝑊 = [(𝑘 − 1) / 𝑁, 𝑘 / 𝑁] is a subset of [0, 1] for each 𝑀 ∈ (1...𝑁). (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (0[,]1)) | ||
| Theorem | cvmliftlem3 35271* | Lemma for cvmlift 35283. Since 1st ‘(𝑇‘𝑀) is a neighborhood of (𝐺 “ 𝑊), every element 𝐴 ∈ 𝑊 satisfies (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀)). (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝐺‘𝐴) ∈ (1st ‘(𝑇‘𝑀))) | ||
| Theorem | cvmliftlem4 35272* | Lemma for cvmlift 35283. The function 𝑄 will be our lifted path, defined piecewise on each section [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] for 𝑀 ∈ (1...𝑁). For 𝑀 = 0, it is a "seed" value which makes the rest of the recursion work, a singleton function mapping 0 to 𝑃. (Contributed by Mario Carneiro, 15-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) ⇒ ⊢ (𝑄‘0) = {⟨0, 𝑃⟩} | ||
| Theorem | cvmliftlem5 35273* | Lemma for cvmlift 35283. Definition of 𝑄 at a successor. This is a function defined on 𝑊 as ◡(𝑇 ↾ 𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2nd ‘(𝑇‘𝑀) that contains 𝑄(𝑀 − 1) evaluated at the last defined point, namely (𝑀 − 1) / 𝑁 (note that for 𝑀 = 1 this is using the seed value 𝑄(0)(0) = 𝑃). (Contributed by Mario Carneiro, 15-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑄‘𝑀) = (𝑧 ∈ 𝑊 ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))) | ||
| Theorem | cvmliftlem6 35274* | Lemma for cvmlift 35283. Induction step for cvmliftlem7 35275. Assuming that 𝑄(𝑀 − 1) is defined at (𝑀 − 1) / 𝑁 and is a preimage of 𝐺((𝑀 − 1) / 𝑁), the next segment 𝑄(𝑀) is also defined and is a function on 𝑊 which is a lift 𝐺 for this segment. This follows explicitly from the definition 𝑄(𝑀) = ◡(𝐹 ↾ 𝐼) ∘ 𝐺 since 𝐺 is in 1st ‘(𝐹‘𝑀) for the entire interval so that ◡(𝐹 ↾ 𝐼) maps this into 𝐼 and 𝐹 ∘ 𝑄 maps back to 𝐺. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ (1...𝑁)) & ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝑄‘𝑀):𝑊⟶𝐵 ∧ (𝐹 ∘ (𝑄‘𝑀)) = (𝐺 ↾ 𝑊))) | ||
| Theorem | cvmliftlem7 35275* | Lemma for cvmlift 35283. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 35274 to show functionality and lifting of 𝑄). (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (◡𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))})) | ||
| Theorem | cvmliftlem8 35276* | Lemma for cvmlift 35283. The functions 𝑄 are continuous functions because they are defined as ◡(𝐹 ↾ 𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹 ↾ 𝐼) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁)) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → (𝑄‘𝑀) ∈ ((𝐿 ↾t 𝑊) Cn 𝐶)) | ||
| Theorem | cvmliftlem9 35277* | Lemma for cvmlift 35283. The 𝑄(𝑀) functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the 𝑄 functions agree on their common domain. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ (1...𝑁)) → ((𝑄‘𝑀)‘((𝑀 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁))) | ||
| Theorem | cvmliftlem10 35278* | Lemma for cvmlift 35283. The function 𝐾 is going to be our complete lifted path, formed by unioning together all the 𝑄 functions (each of which is defined on one segment [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁] of the interval). Here we prove by induction that 𝐾 is a continuous function and a lift of 𝐺 by applying cvmliftlem6 35274, cvmliftlem7 35275 (to show it is a function and a lift), cvmliftlem8 35276 (to show it is continuous), and cvmliftlem9 35277 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 23317 gives that 𝐾 is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) & ⊢ (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ (∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘) ∈ ((𝐿 ↾t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ ∪ 𝑘 ∈ (1...𝑛)(𝑄‘𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁)))))) ⇒ ⊢ (𝜑 → (𝐾 ∈ ((𝐿 ↾t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁))))) | ||
| Theorem | cvmliftlem11 35279* | Lemma for cvmlift 35283. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) ⇒ ⊢ (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = 𝐺)) | ||
| Theorem | cvmliftlem13 35280* | Lemma for cvmlift 35283. The initial value of 𝐾 is 𝑃 because 𝑄(1) is a subset of 𝐾 which takes value 𝑃 at 0. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) ⇒ ⊢ (𝜑 → (𝐾‘0) = 𝑃) | ||
| Theorem | cvmliftlem14 35281* | Lemma for cvmlift 35283. Putting the results of cvmliftlem11 35279, cvmliftlem13 35280 and cvmliftmo 35268 together, we have that 𝐾 is a continuous function, satisfies 𝐹 ∘ 𝐾 = 𝐺 and 𝐾(0) = 𝑃, and is equal to any other function which also has these properties, so it follows that 𝐾 is the unique lift of 𝐺. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇:(1...𝑁)⟶∪ 𝑗 ∈ 𝐽 ({𝑗} × (𝑆‘𝑗))) & ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇‘𝑘))) & ⊢ 𝐿 = (topGen‘ran (,)) & ⊢ 𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ (◡(𝐹 ↾ (℩𝑏 ∈ (2nd ‘(𝑇‘𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺‘𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩})) & ⊢ 𝐾 = ∪ 𝑘 ∈ (1...𝑁)(𝑄‘𝑘) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) | ||
| Theorem | cvmliftlem15 35282* | Lemma for cvmlift 35283. Discharge the assumptions of cvmliftlem14 35281. The set of all open subsets 𝑢 of the unit interval such that 𝐺 “ 𝑢 is contained in an even covering of some open set in 𝐽 is a cover of II by the definition of a covering map, so by the Lebesgue number lemma lebnumii 25011, there is a subdivision of the closed unit interval into 𝑁 equal parts such that each part is entirely contained within one such open set of 𝐽. Then using finite choice ac6sfi 9317 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35281. (Contributed by Mario Carneiro, 14-Feb-2015.) |
| ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) | ||
| Theorem | cvmlift 35283* | One of the important properties of covering maps is that any path 𝐺 in the base space "lifts" to a path 𝑓 in the covering space such that 𝐹 ∘ 𝑓 = 𝐺, and given a starting point 𝑃 in the covering space this lift is unique. The proof is contained in cvmliftlem1 35269 thru cvmliftlem15 35282. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 ⇒ ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) | ||
| Theorem | cvmfo 35284 | A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵–onto→𝑋) | ||
| Theorem | cvmliftiota 35285* | Write out a function 𝐻 that is the unique lift of 𝐹. (Contributed by Mario Carneiro, 16-Feb-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃)) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0)) ⇒ ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃)) | ||
| Theorem | cvmlift2lem1 35286* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ (∀𝑦 ∈ (0[,]1)∃𝑢 ∈ ((nei‘II)‘{𝑦})((𝑢 × {𝑥}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀) → (((0[,]1) × {𝑥}) ⊆ 𝑀 → ((0[,]1) × {𝑡}) ⊆ 𝑀)) | ||
| Theorem | cvmlift2lem9a 35287* | Lemma for cvmlift2 35300 and cvmlift3 35312. (Contributed by Mario Carneiro, 9-Jul-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ 𝑌 = ∪ 𝐾 & ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐻:𝑌⟶𝐵) & ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∈ (𝐾 Cn 𝐽)) & ⊢ (𝜑 → 𝐾 ∈ Top) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) & ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴)) & ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊)) & ⊢ (𝜑 → 𝑀 ⊆ 𝑌) & ⊢ (𝜑 → (𝐻 “ 𝑀) ⊆ 𝑊) ⇒ ⊢ (𝜑 → (𝐻 ↾ 𝑀) ∈ ((𝐾 ↾t 𝑀) Cn 𝐶)) | ||
| Theorem | cvmlift2lem2 35288* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) ⇒ ⊢ (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃)) | ||
| Theorem | cvmlift2lem3 35289* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻‘𝑋))) | ||
| Theorem | cvmlift2lem4 35290* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌)) | ||
| Theorem | cvmlift2lem5 35291* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ (𝜑 → 𝐾:((0[,]1) × (0[,]1))⟶𝐵) | ||
| Theorem | cvmlift2lem6 35292* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×t II) ↾t ({𝑋} × (0[,]1))) Cn 𝐶)) | ||
| Theorem | cvmlift2lem7 35293* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐾) = 𝐺) | ||
| Theorem | cvmlift2lem8 35294* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 9-Mar-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → (𝑋𝐾0) = (𝐻‘𝑋)) | ||
| Theorem | cvmlift2lem9 35295* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) & ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀) & ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝑀)) & ⊢ (𝜑 → 𝑈 ∈ II) & ⊢ (𝜑 → 𝑉 ∈ II) & ⊢ (𝜑 → (II ↾t 𝑈) ∈ Conn) & ⊢ (𝜑 → (II ↾t 𝑉) ∈ Conn) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑈 × 𝑉) ⊆ (◡𝐺 “ 𝑀)) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×t II) ↾t (𝑈 × {𝑍})) Cn 𝐶)) & ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝑋𝐾𝑌) ∈ 𝑏) ⇒ ⊢ (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶)) | ||
| Theorem | cvmlift2lem10 35296* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) & ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) & ⊢ (𝜑 → 𝑋 ∈ (0[,]1)) & ⊢ (𝜑 → 𝑌 ∈ (0[,]1)) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋 ∈ 𝑢 ∧ 𝑌 ∈ 𝑣 ∧ (∃𝑤 ∈ 𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶)))) | ||
| Theorem | cvmlift2lem11 35297* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) & ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)} & ⊢ (𝜑 → 𝑈 ∈ II) & ⊢ (𝜑 → 𝑉 ∈ II) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 (𝐾 ↾ (𝑈 × {𝑤})) ∈ (((II ×t II) ↾t (𝑈 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶))) ⇒ ⊢ (𝜑 → ((𝑈 × {𝑌}) ⊆ 𝑀 → (𝑈 × {𝑍}) ⊆ 𝑀)) | ||
| Theorem | cvmlift2lem12 35298* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) & ⊢ 𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)} & ⊢ 𝐴 = {𝑎 ∈ (0[,]1) ∣ ((0[,]1) × {𝑎}) ⊆ 𝑀} & ⊢ 𝑆 = {⟨𝑟, 𝑡⟩ ∣ (𝑡 ∈ (0[,]1) ∧ ∃𝑢 ∈ ((nei‘II)‘{𝑟})((𝑢 × {𝑎}) ⊆ 𝑀 ↔ (𝑢 × {𝑡}) ⊆ 𝑀))} ⇒ ⊢ (𝜑 → 𝐾 ∈ ((II ×t II) Cn 𝐶)) | ||
| Theorem | cvmlift2lem13 35299* | Lemma for cvmlift2 35300. (Contributed by Mario Carneiro, 7-May-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) & ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) & ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) ⇒ ⊢ (𝜑 → ∃!𝑔 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃)) | ||
| Theorem | cvmlift2 35300* | A two-dimensional version of cvmlift 35283. There is a unique lift of functions on the unit square II ×t II which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.) |
| ⊢ 𝐵 = ∪ 𝐶 & ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) & ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) ⇒ ⊢ (𝜑 → ∃!𝑓 ∈ ((II ×t II) Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (0𝑓0) = 𝑃)) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |