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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcscon 30301 Extend class notation with the class of simply connected topologies.
class SCon

Definitiondf-pcon 30302* Define the class of path-connected topologies. A topology is path-connected if there is a path (a continuous function from the unit interval) that goes from 𝑥 to 𝑦 for any points 𝑥, 𝑦 in the space. (Contributed by Mario Carneiro, 11-Feb-2015.)
PCon = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}

Definitiondf-scon 30303* 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.) (New usage is discouraged.)
SCon = {𝑗 ∈ PCon ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝑗)((0[,]1) × {(𝑓‘0)}))}

Theoremispcon 30304* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽       (𝐽 ∈ PCon ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))

Theorempconcn 30305* The property of being a path-connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑋 = 𝐽       ((𝐽 ∈ PCon ∧ 𝐴𝑋𝐵𝑋) → ∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝐴 ∧ (𝑓‘1) = 𝐵))

Theorempcontop 30306 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ PCon → 𝐽 ∈ Top)

Theoremisscon 30307* The property of being a simply connected topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ SCon ↔ (𝐽 ∈ PCon ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))

Theoremsconpcon 30308 A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ SCon → 𝐽 ∈ PCon)

Theoremscontop 30309 A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ SCon → 𝐽 ∈ Top)

Theoremsconpht 30310 A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
((𝐽 ∈ SCon ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))

Theoremcnpcon 30311 An image of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑌 = 𝐾       ((𝐽 ∈ PCon ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ PCon)

Theorempconcon 30312 A path-connected space is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐽 ∈ PCon → 𝐽 ∈ Con)

Theoremtxpcon 30313 The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×t 𝑆) ∈ PCon)

Theoremptpcon 30314 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.)
((𝐴𝑉𝐹:𝐴⟶PCon) → (∏t𝐹) ∈ PCon)

Theoremindispcon 30315 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} ∈ PCon

Theoremconpcon 30316 A connected and locally path-connected space is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.)
((𝐽 ∈ Con ∧ 𝐽 ∈ 𝑛-Locally PCon) → 𝐽 ∈ PCon)

Theoremqtoppcon 30317 A quotient of a path-connected space is path-connected. (Contributed by Mario Carneiro, 24-Mar-2015.)
𝑋 = 𝐽       ((𝐽 ∈ PCon ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ PCon)

Theorempconpi1 30318 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 ∈ PCon ∧ 𝐴𝑋𝐵𝑋) → 𝑃𝑔 𝑄)

Theoremsconpht2 30319 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.)
(𝜑𝐽 ∈ SCon)    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   (𝜑 → (𝐹‘1) = (𝐺‘1))       (𝜑𝐹( ≃ph𝐽)𝐺)

Theoremsconpi1 30320 A path-connected topological space is simply connected iff its fundamental group is trivial. (Contributed by Mario Carneiro, 12-Feb-2015.)
𝑋 = 𝐽       ((𝐽 ∈ PCon ∧ 𝑌𝑋) → (𝐽 ∈ SCon ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1𝑜))

Theoremtxsconlem 30321 Lemma for txscon 30322. (Contributed by Mario Carneiro, 9-Mar-2015.)
(𝜑𝑅 ∈ Top)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝐹 ∈ (II Cn (𝑅 ×t 𝑆)))    &   𝐴 = ((1st ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)    &   𝐵 = ((2nd ↾ ( 𝑅 × 𝑆)) ∘ 𝐹)    &   (𝜑𝐺 ∈ (𝐴(PHtpy‘𝑅)((0[,]1) × {(𝐴‘0)})))    &   (𝜑𝐻 ∈ (𝐵(PHtpy‘𝑆)((0[,]1) × {(𝐵‘0)})))       (𝜑𝐹( ≃ph‘(𝑅 ×t 𝑆))((0[,]1) × {(𝐹‘0)}))

Theoremtxscon 30322 The topological product of two simply connected spaces is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
((𝑅 ∈ SCon ∧ 𝑆 ∈ SCon) → (𝑅 ×t 𝑆) ∈ SCon)

Theoremcvxpcon 30323* A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑆)       (𝜑𝐾 ∈ PCon)

Theoremcvxscon 30324* A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝑆)       (𝜑𝐾 ∈ SCon)

Theoremblscon 30325 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 = (𝐽t 𝑆)       ((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) → 𝐾 ∈ SCon)

Theoremcnllyscon 30326 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Locally SCon

Theoremrescon 30327 A subset of is simply connected iff it is connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐽 = ((topGen‘ran (,)) ↾t 𝐴)       (𝐴 ⊆ ℝ → (𝐽 ∈ SCon ↔ 𝐽 ∈ Con))

Theoremiooscon 30328 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SCon

Theoremiccscon 30329 A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SCon)

Theoremretopscon 30330 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
(topGen‘ran (,)) ∈ SCon

Theoremiccllyscon 30331 A closed interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Locally SCon)

Theoremrellyscon 30332 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
(topGen‘ran (,)) ∈ Locally SCon

Theoremiiscon 30333 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
II ∈ SCon

Theoremiillyscon 30334 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
II ∈ Locally SCon

Theoremiinllycon 30335 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
II ∈ 𝑛-Locally Con

20.5.8  Covering maps

Syntaxccvm 30336 Extend class notation with the class of covering maps.
class CovMap

Definitiondf-cvm 30337* Define the class of covering maps on two topological spaces. A function 𝑓:𝑐𝑗 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 𝑢𝑠 onto 𝑘. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))))})

Theoremfncvm 30338 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap Fn (Top × Top)

Theoremcvmscbv 30339* Change bound variables in the set of even coverings. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       𝑆 = (𝑎𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑏 = (𝐹𝑎) ∧ ∀𝑐𝑏 (∀𝑑 ∈ (𝑏 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑎))))})

Theoremiscvm 30340* The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝑋 = 𝐽       (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))

Theoremcvmtop1 30341 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

Theoremcvmtop2 30342 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)

Theoremcvmcn 30343 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))

Theoremcvmcov 30344* 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝑋 = 𝐽       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃𝑋) → ∃𝑥𝐽 (𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))

Theoremcvmsrcl 30345* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝑇 ∈ (𝑆𝑈) → 𝑈𝐽)

Theoremcvmsi 30346* One direction of cvmsval 30347. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))

Theoremcvmsval 30347* 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 𝑢𝑆 is homeomorphic under 𝐹 to 𝑈. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝐶𝑉 → (𝑇 ∈ (𝑆𝑈) ↔ (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))

Theoremcvmsss 30348* 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)

Theoremcvmsn0 30349* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝑇 ∈ (𝑆𝑈) → 𝑇 ≠ ∅)

Theoremcvmsuni 30350* An even covering of 𝑈 has union equal to the preimage of 𝑈 by 𝐹. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))

Theoremcvmsdisj 30351* An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))

Theoremcvmshmeo 30352* Every element of an even covering of 𝑈 is homeomorphic to 𝑈 via 𝐹. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴) ∈ ((𝐶t 𝐴)Homeo(𝐽t 𝑈)))

Theoremcvmsf1o 30353* 𝐹, localized to an element of an even covering of 𝑈, is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝐴):𝐴1-1-onto𝑈)

Theoremcvmscld 30354* The sets of an even covering are clopen in the subspace topology on 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 ∈ (Clsd‘(𝐶t (𝐹𝑈))))

Theoremcvmsss2 30355* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑉𝐽𝑉𝑈) → ((𝑆𝑈) ≠ ∅ → (𝑆𝑉) ≠ ∅))

Theoremcvmcov2 30356* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑈𝐽𝑃𝑈) → ∃𝑥 ∈ 𝒫 𝑈(𝑃𝑥 ∧ (𝑆𝑥) ≠ ∅))

Theoremcvmseu 30357* Every element in 𝑇 is a member of a unique element of 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → ∃!𝑥𝑇 𝐴𝑥)

Theoremcvmsiota 30358* Identify the unique element of 𝑇 containing 𝐴. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑊 = (𝑥𝑇 𝐴𝑥)       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝐵 ∧ (𝐹𝐴) ∈ 𝑈)) → (𝑊𝑇𝐴𝑊))

Theoremcvmopnlem 30359* Lemma for cvmopn 30361. (Contributed by Mario Carneiro, 7-May-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶       ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴𝐶) → (𝐹𝐴) ∈ 𝐽)

Theoremcvmfolem 30360* Lemma for cvmfo 30381. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽       (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)

Theoremcvmopn 30361 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴𝐶) → (𝐹𝐴) ∈ 𝐽)

Theoremcvmliftmolem1 30362* Lemma for cvmliftmo 30365. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   ((𝜑𝜓) → 𝑇 ∈ (𝑆𝑈))    &   ((𝜑𝜓) → 𝑊𝑇)    &   ((𝜑𝜓) → 𝐼 ⊆ (𝑀𝑊))    &   ((𝜑𝜓) → (𝐾t 𝐼) ∈ Con)    &   ((𝜑𝜓) → 𝑋𝐼)    &   ((𝜑𝜓) → 𝑄𝐼)    &   ((𝜑𝜓) → 𝑅𝐼)    &   ((𝜑𝜓) → (𝐹‘(𝑀𝑋)) ∈ 𝑈)       ((𝜑𝜓) → (𝑄 ∈ dom (𝑀𝑁) → 𝑅 ∈ dom (𝑀𝑁)))

Theoremcvmliftmolem2 30363* Lemma for cvmliftmo 30365. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝜑𝑀 = 𝑁)

Theoremcvmliftmoi 30364 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    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))       (𝜑𝑀 = 𝑁)

Theoremcvmliftmo 30365* 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    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

Theoremcvmliftlem1 30366* Lemma for cvmlift 30380. In cvmliftlem15 30379, 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 ‘(𝑇𝑀) is a neighborhood of (𝐺 “ [(𝑀 − 1) / 𝑁, 𝑀 / 𝑁]), and 2nd ‘(𝑇𝑀) is an even covering of 1st ‘(𝑇𝑀), which is to say a disjoint union of open sets in 𝐶 whose image is 1st ‘(𝑇𝑀). (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))       ((𝜑𝜓) → (2nd ‘(𝑇𝑀)) ∈ (𝑆‘(1st ‘(𝑇𝑀))))

Theoremcvmliftlem2 30367* Lemma for cvmlift 30380. 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))       ((𝜑𝜓) → 𝑊 ⊆ (0[,]1))

Theoremcvmliftlem3 30368* Lemma for cvmlift 30380. Since 1st ‘(𝑇𝑀) is a neighborhood of (𝐺𝑊), every element 𝐴𝑊 satisfies (𝐺𝐴) ∈ (1st ‘(𝑇𝑀)). (Contributed by Mario Carneiro, 16-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    &   ((𝜑𝜓) → 𝐴𝑊)       ((𝜑𝜓) → (𝐺𝐴) ∈ (1st ‘(𝑇𝑀)))

Theoremcvmliftlem4 30369* Lemma for cvmlift 30380. 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))       (𝑄‘0) = {⟨0, 𝑃⟩}

Theoremcvmliftlem5 30370* Lemma for cvmlift 30380. Definition of 𝑄 at a successor. This is a function defined on 𝑊 as (𝑇𝐼) ∘ 𝐺 where 𝐼 is the unique covering set of 2nd ‘(𝑇𝑀) 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))       ((𝜑𝑀 ∈ ℕ) → (𝑄𝑀) = (𝑧𝑊 ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑀))((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧))))

Theoremcvmliftlem6 30371* Lemma for cvmlift 30380. Induction step for cvmliftlem7 30372. 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) = (𝐹𝐼) ∘ 𝐺 since 𝐺 is in 1st ‘(𝐹𝑀) for the entire interval so that (𝐹𝐼) maps this into 𝐼 and 𝐹𝑄 maps back to 𝐺. (Contributed by Mario Carneiro, 16-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))    &   ((𝜑𝜓) → 𝑀 ∈ (1...𝑁))    &   ((𝜑𝜓) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))       ((𝜑𝜓) → ((𝑄𝑀):𝑊𝐵 ∧ (𝐹 ∘ (𝑄𝑀)) = (𝐺𝑊)))

Theoremcvmliftlem7 30372* Lemma for cvmlift 30380. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 30371 to show functionality and lifting of 𝑄). (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))       ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)) ∈ (𝐹 “ {(𝐺‘((𝑀 − 1) / 𝑁))}))

Theoremcvmliftlem8 30373* Lemma for cvmlift 30380. The functions 𝑄 are continuous functions because they are defined as (𝐹𝐼) ∘ 𝐺 where 𝐺 is continuous and (𝐹𝐼) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝑊 = (((𝑀 − 1) / 𝑁)[,](𝑀 / 𝑁))       ((𝜑𝑀 ∈ (1...𝑁)) → (𝑄𝑀) ∈ ((𝐿t 𝑊) Cn 𝐶))

Theoremcvmliftlem9 30374* Lemma for cvmlift 30380. 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))       ((𝜑𝑀 ∈ (1...𝑁)) → ((𝑄𝑀)‘((𝑀 − 1) / 𝑁)) = ((𝑄‘(𝑀 − 1))‘((𝑀 − 1) / 𝑁)))

Theoremcvmliftlem10 30375* Lemma for cvmlift 30380. 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 30371, cvmliftlem7 30372 (to show it is a function and a lift), cvmliftlem8 30373 (to show it is continuous), and cvmliftlem9 30374 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 20811 gives that 𝐾 is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)    &   (𝜒 ↔ ((𝑛 ∈ ℕ ∧ (𝑛 + 1) ∈ (1...𝑁)) ∧ ( 𝑘 ∈ (1...𝑛)(𝑄𝑘) ∈ ((𝐿t (0[,](𝑛 / 𝑁))) Cn 𝐶) ∧ (𝐹 𝑘 ∈ (1...𝑛)(𝑄𝑘)) = (𝐺 ↾ (0[,](𝑛 / 𝑁))))))       (𝜑 → (𝐾 ∈ ((𝐿t (0[,](𝑁 / 𝑁))) Cn 𝐶) ∧ (𝐹𝐾) = (𝐺 ↾ (0[,](𝑁 / 𝑁)))))

Theoremcvmliftlem11 30376* Lemma for cvmlift 30380. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)       (𝜑 → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹𝐾) = 𝐺))

Theoremcvmliftlem13 30377* Lemma for cvmlift 30380. 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)       (𝜑 → (𝐾‘0) = 𝑃)

Theoremcvmliftlem14 30378* Lemma for cvmlift 30380. Putting the results of cvmliftlem11 30376, cvmliftlem13 30377 and cvmliftmo 30365 together, we have that 𝐾 is a continuous function, satisfies 𝐹𝐾 = 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇:(1...𝑁)⟶ 𝑗𝐽 ({𝑗} × (𝑆𝑗)))    &   (𝜑 → ∀𝑘 ∈ (1...𝑁)(𝐺 “ (((𝑘 − 1) / 𝑁)[,](𝑘 / 𝑁))) ⊆ (1st ‘(𝑇𝑘)))    &   𝐿 = (topGen‘ran (,))    &   𝑄 = seq0((𝑥 ∈ V, 𝑚 ∈ ℕ ↦ (𝑧 ∈ (((𝑚 − 1) / 𝑁)[,](𝑚 / 𝑁)) ↦ ((𝐹 ↾ (𝑏 ∈ (2nd ‘(𝑇𝑚))(𝑥‘((𝑚 − 1) / 𝑁)) ∈ 𝑏))‘(𝐺𝑧)))), (( I ↾ ℕ) ∪ {⟨0, {⟨0, 𝑃⟩}⟩}))    &   𝐾 = 𝑘 ∈ (1...𝑁)(𝑄𝑘)       (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Theoremcvmliftlem15 30379* Lemma for cvmlift 30380. Discharge the assumptions of cvmliftlem14 30378. The set of all open subsets 𝑢 of the unit interval such that 𝐺𝑢 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 22496, there is a subdivision of the 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 7965 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 30378. (Contributed by Mario Carneiro, 14-Feb-2015.)
𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   𝐵 = 𝐶    &   𝑋 = 𝐽    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))       (𝜑 → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Theoremcvmlift 30380* 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 𝐹𝑓 = 𝐺, and given a starting point 𝑃 in the covering space this lift is unique. The proof is contained in cvmliftlem1 30366 thru cvmliftlem15 30379. (Contributed by Mario Carneiro, 16-Feb-2015.)
𝐵 = 𝐶       (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽)) ∧ (𝑃𝐵 ∧ (𝐹𝑃) = (𝐺‘0))) → ∃!𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))

Theoremcvmfo 30381 A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
𝐵 = 𝐶    &   𝑋 = 𝐽       (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹:𝐵onto𝑋)

Theoremcvmliftiota 30382* 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)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))       (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹𝐻) = 𝐺 ∧ (𝐻‘0) = 𝑃))

Theoremcvmlift2lem1 30383* Lemma for cvmlift2 30397. (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))

Theoremcvmlift2lem9a 30384* Lemma for cvmlift2 30397 and cvmlift3 30409. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐻:𝑌𝐵)    &   (𝜑 → (𝐹𝐻) ∈ (𝐾 Cn 𝐽))    &   (𝜑𝐾 ∈ Top)    &   (𝜑𝑋𝑌)    &   (𝜑𝑇 ∈ (𝑆𝐴))    &   (𝜑 → (𝑊𝑇 ∧ (𝐻𝑋) ∈ 𝑊))    &   (𝜑𝑀𝑌)    &   (𝜑 → (𝐻𝑀) ⊆ 𝑊)       (𝜑 → (𝐻𝑀) ∈ ((𝐾t 𝑀) Cn 𝐶))

Theoremcvmlift2lem2 30385* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))       (𝜑 → (𝐻 ∈ (II Cn 𝐶) ∧ (𝐹𝐻) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝐻‘0) = 𝑃))

Theoremcvmlift2lem3 30386* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑋)))       ((𝜑𝑋 ∈ (0[,]1)) → (𝐾 ∈ (II Cn 𝐶) ∧ (𝐹𝐾) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝐾‘0) = (𝐻𝑋)))

Theoremcvmlift2lem4 30387* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑋)))‘𝑌))

Theoremcvmlift2lem5 30388* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       (𝜑𝐾:((0[,]1) × (0[,]1))⟶𝐵)

Theoremcvmlift2lem6 30389* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       ((𝜑𝑋 ∈ (0[,]1)) → (𝐾 ↾ ({𝑋} × (0[,]1))) ∈ (((II ×t II) ↾t ({𝑋} × (0[,]1))) Cn 𝐶))

Theoremcvmlift2lem7 30390* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       (𝜑 → (𝐹𝐾) = 𝐺)

Theoremcvmlift2lem8 30391* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       ((𝜑𝑋 ∈ (0[,]1)) → (𝑋𝐾0) = (𝐻𝑋))

Theoremcvmlift2lem9 30392* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)    &   (𝜑𝑇 ∈ (𝑆𝑀))    &   (𝜑𝑈 ∈ II)    &   (𝜑𝑉 ∈ II)    &   (𝜑 → (II ↾t 𝑈) ∈ Con)    &   (𝜑 → (II ↾t 𝑉) ∈ Con)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑈 × 𝑉) ⊆ (𝐺𝑀))    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×t II) ↾t (𝑈 × {𝑍})) Cn 𝐶))    &   𝑊 = (𝑏𝑇 (𝑋𝐾𝑌) ∈ 𝑏)       (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶))

Theoremcvmlift2lem10 30393* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑𝑋 ∈ (0[,]1))    &   (𝜑𝑌 ∈ (0[,]1))       (𝜑 → ∃𝑢 ∈ II ∃𝑣 ∈ II (𝑋𝑢𝑌𝑣 ∧ (∃𝑤𝑣 (𝐾 ↾ (𝑢 × {𝑤})) ∈ (((II ×t II) ↾t (𝑢 × {𝑤})) Cn 𝐶) → (𝐾 ↾ (𝑢 × 𝑣)) ∈ (((II ×t II) ↾t (𝑢 × 𝑣)) Cn 𝐶))))

Theoremcvmlift2lem11 30394* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))    &   𝑀 = {𝑧 ∈ ((0[,]1) × (0[,]1)) ∣ 𝐾 ∈ (((II ×t II) CnP 𝐶)‘𝑧)}    &   (𝜑𝑈 ∈ II)    &   (𝜑𝑉 ∈ II)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → (∃𝑤𝑉 (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))

Theoremcvmlift2lem12 30395* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘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))}       (𝜑𝐾 ∈ ((II ×t II) Cn 𝐶))

Theoremcvmlift2lem13 30396* Lemma for cvmlift2 30397. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))       (𝜑 → ∃!𝑔 ∈ ((II ×t II) Cn 𝐶)((𝐹𝑔) = 𝐺 ∧ (0𝑔0) = 𝑃))

Theoremcvmlift2 30397* A two-dimensional version of cvmlift 30380. 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    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))       (𝜑 → ∃!𝑓 ∈ ((II ×t II) Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (0𝑓0) = 𝑃))

Theoremcvmliftphtlem 30398* Lemma for cvmliftpht 30399. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑀 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   𝑁 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (II Cn 𝐽))    &   (𝜑𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))    &   (𝜑𝐴 ∈ ((II ×t II) Cn 𝐶))    &   (𝜑 → (𝐹𝐴) = 𝐾)    &   (𝜑 → (0𝐴0) = 𝑃)       (𝜑𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))

Theoremcvmliftpht 30399* If 𝐺 and 𝐻 are path-homotopic, then their lifts 𝑀 and 𝑁 are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑀 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   𝑁 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺‘0))    &   (𝜑𝐺( ≃ph𝐽)𝐻)       (𝜑𝑀( ≃ph𝐶)𝑁)

Theoremcvmlift3lem1 30400* Lemma for cvmlift3 30409. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   (𝜑𝑀 ∈ (II Cn 𝐾))    &   (𝜑 → (𝑀‘0) = 𝑂)    &   (𝜑𝑁 ∈ (II Cn 𝐾))    &   (𝜑 → (𝑁‘0) = 𝑂)    &   (𝜑 → (𝑀‘1) = (𝑁‘1))       (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = 𝑃))‘1))

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