P. 354 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	qtoppconn 35301	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 35302	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 35303	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 35304	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 35305	Lemma for txsconn 35306. (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 35306	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 35307*	A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11093. (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 35308*	A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11093. (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 35309	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 35310	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 35311	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 35312	An open interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ ((topGen‘ran (,)) ↾t (𝐴(,)𝐵)) ∈ SConn
Theorem	iccsconn 35313	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 35314	The real numbers are simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ SConn
Theorem	iccllysconn 35315	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 35316	The real numbers are locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ Locally SConn
Theorem	iisconn 35317	The unit interval is simply connected. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ II ∈ SConn
Theorem	iillysconn 35318	The unit interval is locally simply connected. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ II ∈ Locally SConn
Theorem	iinllyconn 35319	The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ II ∈ 𝑛-Locally Conn
21.6.8  Covering maps
Syntax	ccvm 35320	Extend class notation with the class of covering maps.
class CovMap
Definition	df-cvm 35321*	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 35322	Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ CovMap Fn (Top × Top)
Theorem	cvmscbv 35323*	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 35324*	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 35325	Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
Theorem	cvmtop2 35326	Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top)
Theorem	cvmcn 35327	A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
Theorem	cvmcov 35328*	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 35329*	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 35330*	One direction of cvmsval 35331. (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 35331*	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 35332*	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 35333*	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 35334*	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 35335*	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 35336*	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 35337*	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 35338*	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 35339*	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 35340*	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 35341*	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 35342*	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 35343*	Lemma for cvmopn 35345. (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 35344*	Lemma for cvmfo 35365. (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 35345	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 35346*	Lemma for cvmliftmo 35349. (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 35347*	Lemma for cvmliftmo 35349. (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 35348	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 35349*	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 35350*	Lemma for cvmlift 35364. In cvmliftlem15 35363, 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 35351*	Lemma for cvmlift 35364. 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 35352*	Lemma for cvmlift 35364. 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 35353*	Lemma for cvmlift 35364. 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 35354*	Lemma for cvmlift 35364. 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 35355*	Lemma for cvmlift 35364. Induction step for cvmliftlem7 35356. 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 35356*	Lemma for cvmlift 35364. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 35355 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 35357*	Lemma for cvmlift 35364. 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 35358*	Lemma for cvmlift 35364. 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 35359*	Lemma for cvmlift 35364. 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 35355, cvmliftlem7 35356 (to show it is a function and a lift), cvmliftlem8 35357 (to show it is continuous), and cvmliftlem9 35358 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 23210 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 35360*	Lemma for cvmlift 35364. (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 35361*	Lemma for cvmlift 35364. 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 35362*	Lemma for cvmlift 35364. Putting the results of cvmliftlem11 35360, cvmliftlem13 35361 and cvmliftmo 35349 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 35363*	Lemma for cvmlift 35364. Discharge the assumptions of cvmliftlem14 35362. 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 24893, 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 9175 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 35362. (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 35364*	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 35350 thru cvmliftlem15 35363. (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 35365	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 35366*	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 35367*	Lemma for cvmlift2 35381. (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 35368*	Lemma for cvmlift2 35381 and cvmlift3 35393. (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 35369*	Lemma for cvmlift2 35381. (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 35370*	Lemma for cvmlift2 35381. (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 35371*	Lemma for cvmlift2 35381. (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 35372*	Lemma for cvmlift2 35381. (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 35373*	Lemma for cvmlift2 35381. (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 35374*	Lemma for cvmlift2 35381. (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 35375*	Lemma for cvmlift2 35381. (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 35376*	Lemma for cvmlift2 35381. (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 35377*	Lemma for cvmlift2 35381. (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 35378*	Lemma for cvmlift2 35381. (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 35379*	Lemma for cvmlift2 35381. (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 35380*	Lemma for cvmlift2 35381. (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 35381*	A two-dimensional version of cvmlift 35364. 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))
Theorem	cvmliftphtlem 35382*	Lemma for cvmliftpht 35383. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑀 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐻))    &   ⊢ (𝜑 → 𝐴 ∈ ((II ×t II) Cn 𝐶))    &   ⊢ (𝜑 → (𝐹 ∘ 𝐴) = 𝐾)    &   ⊢ (𝜑 → (0𝐴0) = 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑀(PHtpy‘𝐶)𝑁))
Theorem	cvmliftpht 35383*	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)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘0) = 𝑃))    &   ⊢ 𝑁 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐻 ∧ (𝑓‘0) = 𝑃))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐻)    ⇒   ⊢ (𝜑 → 𝑀( ≃ph‘𝐶)𝑁)
Theorem	cvmlift3lem1 35384*	Lemma for cvmlift3 35393. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ (𝜑 → 𝑀 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑀‘0) = 𝑂)    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))    &   ⊢ (𝜑 → (𝑁‘0) = 𝑂)    &   ⊢ (𝜑 → (𝑀‘1) = (𝑁‘1))    ⇒   ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = 𝑃))‘1))
Theorem	cvmlift3lem2 35385*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ∃!𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
Theorem	cvmlift3lem3 35386*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
Theorem	cvmlift3lem4 35387*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑌) → ((𝐻‘𝑋) = 𝐴 ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝐴)))
Theorem	cvmlift3lem5 35388*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
Theorem	cvmlift3lem6 35389*	Lemma for cvmlift3 35393. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 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 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ (II Cn 𝐾))    &   ⊢ 𝑅 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑄) ∧ (𝑔‘0) = 𝑃))    &   ⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))    &   ⊢ (𝜑 → 𝑁 ∈ (II Cn (𝐾 ↾t 𝑀)))    &   ⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))    &   ⊢ 𝐼 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = (𝐻‘𝑋)))    ⇒   ⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)
Theorem	cvmlift3lem7 35390*	Lemma for cvmlift3 35393. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 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 → (𝐺‘𝑋) ∈ 𝐴)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))    &   ⊢ (𝜑 → 𝑀 ⊆ (◡𝐺 “ 𝐴))    &   ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)    &   ⊢ (𝜑 → (𝐾 ↾t 𝑀) ∈ PConn)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))
Theorem	cvmlift3lem8 35391*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 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 → 𝐻 ∈ (𝐾 Cn 𝐶))
Theorem	cvmlift3lem9 35392*	Lemma for cvmlift2 35381. (Contributed by Mario Carneiro, 7-May-2015.)
⊢ 𝐵 = ∪ 𝐶    &   ⊢ 𝑌 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))    &   ⊢ 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 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
Theorem	cvmlift3 35393*	A general version of cvmlift 35364. 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    &   ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ SConn)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)    &   ⊢ (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))
21.6.9  Normal numbers
Theorem	snmlff 35394*	The function 𝐹 from snmlval 35396 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 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ 𝐹:ℕ⟶(0[,]1)
Theorem	snmlfval 35395*	The function 𝐹 from snmlval 35396 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 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) = ((♯‘{𝑘 ∈ (1...𝑁) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑁))
Theorem	snmlval 35396*	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 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    ⇒   ⊢ (𝐴 ∈ (𝑆‘𝑅) ↔ (𝑅 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℝ ∧ ∀𝑏 ∈ (0...(𝑅 − 1))(𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑅)))
Theorem	snmlflim 35397*	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 · (𝑟↑𝑘)) mod 𝑟)) = 𝑏}) / 𝑛)) ⇝ (1 / 𝑟)})    &   ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((♯‘{𝑘 ∈ (1...𝑛) ∣ (⌊‘((𝐴 · (𝑅↑𝑘)) mod 𝑅)) = 𝐵}) / 𝑛))    ⇒   ⊢ ((𝐴 ∈ (𝑆‘𝑅) ∧ 𝐵 ∈ (0...(𝑅 − 1))) → 𝐹 ⇝ (1 / 𝑅))
21.6.10  Godel-sets of formulas - part 1
Syntax	cgoe 35398	The Godel-set of membership.
class ∈𝑔
Syntax	cgna 35399	The Godel-set for the Sheffer stroke.
class ⊼𝑔
Syntax	cgol 35400	The Godel-set of universal quantification. (Note that this is not a wff.)
class ∀𝑔𝑁𝑈