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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremindispconn 31201 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

Theoremconnpconn 31202 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)

Theoremqtoppconn 31203 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)

Theorempconnpi1 31204 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 ∧ 𝐴𝑋𝐵𝑋) → 𝑃𝑔 𝑄)

Theoremsconnpht2 31205 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 ∈ SConn)    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   (𝜑 → (𝐹‘1) = (𝐺‘1))       (𝜑𝐹( ≃ph𝐽)𝐺)

Theoremsconnpi1 31206 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 ∧ 𝑌𝑋) → (𝐽 ∈ SConn ↔ (Base‘(𝐽 π1 𝑌)) ≈ 1𝑜))

Theoremtxsconnlem 31207 Lemma for txsconn 31208. (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)}))

Theoremtxsconn 31208 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)

Theoremcvxpconn 31209* 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 ∈ PConn)

Theoremcvxsconn 31210* 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 ∈ SConn)

Theoremblsconn 31211 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 ∈ SConn)

Theoremcnllysconn 31212 The topology of the complex numbers is locally simply connected. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Locally SConn

Theoremresconn 31213 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 ∈ SConn ↔ 𝐽 ∈ Conn))

Theoremioosconn 31214 An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn

Theoremiccsconn 31215 A closed interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ SConn)

Theoremretopsconn 31216 The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
(topGen‘ran (,)) ∈ SConn

Theoremiccllysconn 31217 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)

Theoremrellysconn 31218 The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
(topGen‘ran (,)) ∈ Locally SConn

Theoremiisconn 31219 The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
II ∈ SConn

Theoremiillysconn 31220 The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
II ∈ Locally SConn

Theoremiinllyconn 31221 The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
II ∈ 𝑛-Locally Conn

20.5.8  Covering maps

Syntaxccvm 31222 Extend class notation with the class of covering maps.
class CovMap

Definitiondf-cvm 31223* 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 31224 Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap Fn (Top × Top)

Theoremcvmscbv 31225* 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 31226* 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 31227 Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)

Theoremcvmtop2 31228 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)

Theoremcvmcn 31229 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
(𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))

Theoremcvmcov 31230* 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 31231* 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 31232* One direction of cvmsval 31233. (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 31233* 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 31234* 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 31235* 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 31236* 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 31237* 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 31238* 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 31239* 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 31240* 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 31241* 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 31242* 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 31243* 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 31244* 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 31245* Lemma for cvmopn 31247. (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 31246* Lemma for cvmfo 31267. (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 31247 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴𝐶) → (𝐹𝐴) ∈ 𝐽)

Theoremcvmliftmolem1 31248* Lemma for cvmliftmo 31251. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Conn)    &   (𝜑𝐾 ∈ 𝑛-Locally Conn)    &   (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) ∈ Conn)    &   ((𝜑𝜓) → 𝑋𝐼)    &   ((𝜑𝜓) → 𝑄𝐼)    &   ((𝜑𝜓) → 𝑅𝐼)    &   ((𝜑𝜓) → (𝐹‘(𝑀𝑋)) ∈ 𝑈)       ((𝜑𝜓) → (𝑄 ∈ dom (𝑀𝑁) → 𝑅 ∈ dom (𝑀𝑁)))

Theoremcvmliftmolem2 31249* Lemma for cvmliftmo 31251. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Conn)    &   (𝜑𝐾 ∈ 𝑛-Locally Conn)    &   (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 31250 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 ∈ Conn)    &   (𝜑𝐾 ∈ 𝑛-Locally Conn)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))       (𝜑𝑀 = 𝑁)

Theoremcvmliftmo 31251* 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 ∈ Conn)    &   (𝜑𝐾 ∈ 𝑛-Locally Conn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

Theoremcvmliftlem1 31252* Lemma for cvmlift 31266. In cvmliftlem15 31265, 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 31253* Lemma for cvmlift 31266. 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 31254* Lemma for cvmlift 31266. 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 31255* Lemma for cvmlift 31266. 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 31256* Lemma for cvmlift 31266. 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 31257* Lemma for cvmlift 31266. Induction step for cvmliftlem7 31258. 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 31258* Lemma for cvmlift 31266. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 31257 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 31259* Lemma for cvmlift 31266. 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 31260* Lemma for cvmlift 31266. 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 31261* Lemma for cvmlift 31266. 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 31257, cvmliftlem7 31258 (to show it is a function and a lift), cvmliftlem8 31259 (to show it is continuous), and cvmliftlem9 31260 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 21092 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 31262* Lemma for cvmlift 31266. (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 31263* Lemma for cvmlift 31266. 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 31264* Lemma for cvmlift 31266. Putting the results of cvmliftlem11 31262, cvmliftlem13 31263 and cvmliftmo 31251 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 31265* Lemma for cvmlift 31266. Discharge the assumptions of cvmliftlem14 31264. 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 22759, 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 8201 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 31264. (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 31266* 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 31252 thru cvmliftlem15 31265. (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 31267 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 31268* 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 31269* Lemma for cvmlift2 31283. (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 31270* Lemma for cvmlift2 31283 and cvmlift3 31295. (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 31271* Lemma for cvmlift2 31283. (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 31272* Lemma for cvmlift2 31283. (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 31273* Lemma for cvmlift2 31283. (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 31274* Lemma for cvmlift2 31283. (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 31275* Lemma for cvmlift2 31283. (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 31276* Lemma for cvmlift2 31283. (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 31277* Lemma for cvmlift2 31283. (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 31278* Lemma for cvmlift2 31283. (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) ∈ Conn)    &   (𝜑 → (II ↾t 𝑉) ∈ Conn)    &   (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 31279* Lemma for cvmlift2 31283. (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 31280* Lemma for cvmlift2 31283. (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 31281* Lemma for cvmlift2 31283. (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 31282* Lemma for cvmlift2 31283. (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 31283* A two-dimensional version of cvmlift 31266. 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 31284* Lemma for cvmliftpht 31285. (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 31285* 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 31286* Lemma for cvmlift3 31295. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (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))

Theoremcvmlift3lem2 31287* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))

Theoremcvmlift3lem3 31288* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       (𝜑𝐻:𝑌𝐵)

Theoremcvmlift3lem4 31289* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       ((𝜑𝑋𝑌) → ((𝐻𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))

Theoremcvmlift3lem5 31290* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))       (𝜑 → (𝐹𝐻) = 𝐺)

Theoremcvmlift3lem6 31291* Lemma for cvmlift3 31295. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝐺𝑋) ∈ 𝐴)    &   (𝜑𝑇 ∈ (𝑆𝐴))    &   (𝜑𝑀 ⊆ (𝐺𝐴))    &   𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)    &   (𝜑𝑋𝑀)    &   (𝜑𝑍𝑀)    &   (𝜑𝑄 ∈ (II Cn 𝐾))    &   𝑅 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑄) ∧ (𝑔‘0) = 𝑃))    &   (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻𝑋)))    &   (𝜑𝑁 ∈ (II Cn (𝐾t 𝑀)))    &   (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))    &   𝐼 = (𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = (𝐻𝑋)))       (𝜑 → (𝐻𝑍) ∈ 𝑊)

Theoremcvmlift3lem7 31292* Lemma for cvmlift3 31295. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝐺𝑋) ∈ 𝐴)    &   (𝜑𝑇 ∈ (𝑆𝐴))    &   (𝜑𝑀 ⊆ (𝐺𝐴))    &   𝑊 = (𝑏𝑇 (𝐻𝑋) ∈ 𝑏)    &   (𝜑 → (𝐾t 𝑀) ∈ PConn)    &   (𝜑𝑉𝐾)    &   (𝜑𝑉𝑀)    &   (𝜑𝑋𝑉)       (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Theoremcvmlift3lem8 31293* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})       (𝜑𝐻 ∈ (𝐾 Cn 𝐶))

Theoremcvmlift3lem9 31294* Lemma for cvmlift2 31283. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   𝐻 = (𝑥𝑌 ↦ (𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})       (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

Theoremcvmlift3 31295* A general version of cvmlift 31266. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SConn)    &   (𝜑𝐾 ∈ 𝑛-Locally PConn)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

20.5.9  Normal numbers

Theoremsnmlff 31296* The function 𝐹 from snmlval 31298 is a mapping from positive integers to real numbers in the range [0, 1]. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       𝐹:ℕ⟶(0[,]1)

Theoremsnmlfval 31297* The function 𝐹 from snmlval 31298 maps 𝑁 to the relative density of 𝐵 in the first 𝑁 digits of the digit string of 𝐴 in base 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       (𝑁 ∈ ℕ → (𝐹𝑁) = ((#‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))

Theoremsnmlval 31298* The property "𝐴 is simply normal in base 𝑅". A number is simply normal if each digit 0 ≤ 𝑏 < 𝑅 occurs in the base- 𝑅 digit string of 𝐴 with frequency 1 / 𝑅 (which is consistent with the expectation in an infinite random string of numbers selected from 0...𝑅 − 1). (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑆 = (𝑟 ∈ (ℤ‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})       (𝐴 ∈ (𝑆𝑅) ↔ (𝑅 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))

Theoremsnmlflim 31299* If 𝐴 is simply normal, then the function 𝐹 of relative density of 𝐵 in the digit string converges to 1 / 𝑅, i.e. the set of occurrences of 𝐵 in the digit string has natural density 1 / 𝑅. (Contributed by Mario Carneiro, 6-Apr-2015.)
𝑆 = (𝑟 ∈ (ℤ‘2) ↦ {𝑥 ∈ ℝ ∣ ∀𝑏 ∈ (0...(𝑟 − 1))(𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝑥 · (𝑟𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    &   𝐹 = (𝑛 ∈ ℕ ↦ ((#‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))       ((𝐴 ∈ (𝑆𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))

20.5.10  Godel-sets of formulas

Syntaxcgoe 31300 The Godel-set of membership.
class 𝑔

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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