Home Metamath Proof ExplorerTheorem List (p. 306 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

Theorem List for Metamath Proof Explorer - 30501-30600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvmsi 30501* One direction of cvmsval 30502. (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 30502* 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 30503* 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 30504* 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 30505* 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 30506* 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 30507* 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 30508* 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 30509* 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 30510* 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 30511* 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 30512* 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 30513* 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 30514* Lemma for cvmopn 30516. (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 30515* Lemma for cvmfo 30536. (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 30516 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝐴𝐶) → (𝐹𝐴) ∈ 𝐽)

Theoremcvmliftmolem1 30517* Lemma for cvmliftmo 30520. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})    &   ((𝜑𝜓) → 𝑇 ∈ (𝑆𝑈))    &   ((𝜑𝜓) → 𝑊𝑇)    &   ((𝜑𝜓) → 𝐼 ⊆ (𝑀𝑊))    &   ((𝜑𝜓) → (𝐾t 𝐼) ∈ Con)    &   ((𝜑𝜓) → 𝑋𝐼)    &   ((𝜑𝜓) → 𝑄𝐼)    &   ((𝜑𝜓) → 𝑅𝐼)    &   ((𝜑𝜓) → (𝐹‘(𝑀𝑋)) ∈ 𝑈)       ((𝜑𝜓) → (𝑄 ∈ dom (𝑀𝑁) → 𝑅 ∈ dom (𝑀𝑁)))

Theoremcvmliftmolem2 30518* Lemma for cvmliftmo 30520. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})       (𝜑𝑀 = 𝑁)

Theoremcvmliftmoi 30519 A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝑀 ∈ (𝐾 Cn 𝐶))    &   (𝜑𝑁 ∈ (𝐾 Cn 𝐶))    &   (𝜑 → (𝐹𝑀) = (𝐹𝑁))    &   (𝜑 → (𝑀𝑂) = (𝑁𝑂))       (𝜑𝑀 = 𝑁)

Theoremcvmliftmo 30520* A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ Con)    &   (𝜑𝐾 ∈ 𝑛-Locally Con)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

Theoremcvmliftlem1 30521* Lemma for cvmlift 30535. In cvmliftlem15 30534, 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 30522* Lemma for cvmlift 30535. 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 30523* Lemma for cvmlift 30535. 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 30524* Lemma for cvmlift 30535. 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 30525* Lemma for cvmlift 30535. 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 30526* Lemma for cvmlift 30535. Induction step for cvmliftlem7 30527. 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 30527* Lemma for cvmlift 30535. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 30526 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 30528* Lemma for cvmlift 30535. 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 30529* Lemma for cvmlift 30535. 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 30530* Lemma for cvmlift 30535. 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 30526, cvmliftlem7 30527 (to show it is a function and a lift), cvmliftlem8 30528 (to show it is continuous), and cvmliftlem9 30529 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 20908 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 30531* Lemma for cvmlift 30535. (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 30532* Lemma for cvmlift 30535. 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 30533* Lemma for cvmlift 30535. Putting the results of cvmliftlem11 30531, cvmliftlem13 30532 and cvmliftmo 30520 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 30534* Lemma for cvmlift 30535. Discharge the assumptions of cvmliftlem14 30533. 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 22573, 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 8089 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 30533. (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 30535* 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 30521 thru cvmliftlem15 30534. (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 30536 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 30537* 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 30538* Lemma for cvmlift2 30552. (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 30539* Lemma for cvmlift2 30552 and cvmlift3 30564. (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 30540* Lemma for cvmlift2 30552. (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 30541* Lemma for cvmlift2 30552. (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 30542* Lemma for cvmlift2 30552. (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 30543* Lemma for cvmlift2 30552. (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 30544* Lemma for cvmlift2 30552. (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 30545* Lemma for cvmlift2 30552. (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 30546* Lemma for cvmlift2 30552. (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 30547* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 1-Jun-2015.)
𝐵 = 𝐶    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐺 ∈ ((II ×t II) Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (0𝐺0))    &   𝐻 = (𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((𝑓 ∈ (II Cn 𝐶)((𝐹𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻𝑥)))‘𝑦))    &   𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑐𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐𝑑) = ∅ ∧ (𝐹𝑐) ∈ ((𝐶t 𝑐)Homeo(𝐽t 𝑘))))})    &   (𝜑 → (𝑋𝐺𝑌) ∈ 𝑀)    &   (𝜑𝑇 ∈ (𝑆𝑀))    &   (𝜑𝑈 ∈ II)    &   (𝜑𝑉 ∈ II)    &   (𝜑 → (II ↾t 𝑈) ∈ Con)    &   (𝜑 → (II ↾t 𝑉) ∈ Con)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑈 × 𝑉) ⊆ (𝐺𝑀))    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝐾 ↾ (𝑈 × {𝑍})) ∈ (((II ×t II) ↾t (𝑈 × {𝑍})) Cn 𝐶))    &   𝑊 = (𝑏𝑇 (𝑋𝐾𝑌) ∈ 𝑏)       (𝜑 → (𝐾 ↾ (𝑈 × 𝑉)) ∈ (((II ×t II) ↾t (𝑈 × 𝑉)) Cn 𝐶))

Theoremcvmlift2lem10 30548* Lemma for cvmlift2 30552. (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 30549* Lemma for cvmlift2 30552. (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 30550* Lemma for cvmlift2 30552. (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 30551* Lemma for cvmlift2 30552. (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 30552* A two-dimensional version of cvmlift 30535. 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 30553* Lemma for cvmliftpht 30554. (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 30554* 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 30555* Lemma for cvmlift3 30564. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   (𝜑𝑀 ∈ (II Cn 𝐾))    &   (𝜑 → (𝑀‘0) = 𝑂)    &   (𝜑𝑁 ∈ (II Cn 𝐾))    &   (𝜑 → (𝑁‘0) = 𝑂)    &   (𝜑 → (𝑀‘1) = (𝑁‘1))       (𝜑 → ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑀) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑁) ∧ (𝑔‘0) = 𝑃))‘1))

Theoremcvmlift3lem2 30556* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       ((𝜑𝑋𝑌) → ∃!𝑧𝐵𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑋 ∧ ((𝑔 ∈ (II Cn 𝐶)((𝐹𝑔) = (𝐺𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))

Theoremcvmlift3lem3 30557* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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 30558* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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 30559* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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 30560* Lemma for cvmlift3 30564. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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 30561* Lemma for cvmlift3 30564. (Contributed by Mario Carneiro, 9-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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) ∈ PCon)    &   (𝜑𝑉𝐾)    &   (𝜑𝑉𝑀)    &   (𝜑𝑋𝑉)       (𝜑𝐻 ∈ ((𝐾 CnP 𝐶)‘𝑋))

Theoremcvmlift3lem8 30562* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 6-Jul-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))    &   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 30563* Lemma for cvmlift2 30552. (Contributed by Mario Carneiro, 7-May-2015.)
𝐵 = 𝐶    &   𝑌 = 𝐾    &   (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))    &   (𝜑𝐾 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (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 30564* A general version of cvmlift 30535. 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 ∈ SCon)    &   (𝜑𝐾 ∈ 𝑛-Locally PCon)    &   (𝜑𝑂𝑌)    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐽))    &   (𝜑𝑃𝐵)    &   (𝜑 → (𝐹𝑃) = (𝐺𝑂))       (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹𝑓) = 𝐺 ∧ (𝑓𝑂) = 𝑃))

21.5.9  Normal numbers

Theoremsnmlff 30565* The function 𝐹 from snmlval 30567 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 30566* The function 𝐹 from snmlval 30567 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 30567* 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 30568* 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))

21.5.10  Godel-sets of formulas

Syntaxcgoe 30569 The Godel-set of membership.
class 𝑔

Syntaxcgna 30570 The Godel-set for the Sheffer stroke.
class 𝑔

Syntaxcgol 30571 The Godel-set of universal quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈

Syntaxcsat 30572 The satisfaction function.
class Sat

Syntaxcfmla 30573 The formula set predicate.
class Fmla

Syntaxcsate 30574 The -satisfaction function.
class Sat

Syntaxcprv 30575 The "proves" relation.
class

Definitiondf-goel 30576 Define the Godel-set of membership. Here the arguments 𝑥 = ⟨𝑁, 𝑃 correspond to vN and vP , so (∅∈𝑔1𝑜) actually means v0 v1 , not 0 ∈ 1. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑥 ∈ (ω × ω) ↦ ⟨∅, 𝑥⟩)

Definitiondf-gona 30577 Define the Godel-set for the Sheffer stroke NAND. Here the arguments 𝑥 = ⟨𝑈, 𝑉 are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑥 ∈ (V × V) ↦ ⟨1𝑜, 𝑥⟩)

Definitiondf-goal 30578 Define the Godel-set of universal quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∀𝑥𝜑] = ∀𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔𝑁𝑈 = ⟨2𝑜, ⟨𝑁, 𝑈⟩⟩

Definitiondf-sat 30579* Define the satisfaction predicate. This recursive construction builds up a function over wff codes and simultaneously defines the set of assignments to all variables from 𝑀 that makes the coded wff true in the model 𝑀, where is interpreted as the binary relation 𝐸 on 𝑀. The interpretation of the statement 𝑆 ∈ (((𝑀 Sat 𝐸)‘𝑛)‘𝑈) is that for the model 𝑀, 𝐸, 𝑆:ω⟶𝑀 is a valuation of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1𝑜), etc.) and 𝑈 is a code for a wff using ∈ , ⊼ , ∀ that is true under the assignment 𝑆. The function is defined by finite recursion; ((𝑀 Sat 𝐸)‘𝑛) only operates on wffs of depth at most 𝑛 ∈ ω, and ((𝑀 Sat 𝐸)‘ω) = 𝑛 ∈ ω((𝑀 Sat 𝐸)‘𝑛) operates on all wffs. The coding scheme for the wffs is defined so that
• vi vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩,
• (𝜑𝜓) is coded as ⟨1𝑜, ⟨𝜑, 𝜓⟩⟩, and
• vi 𝜑 is coded as ⟨2𝑜, ⟨𝑖, 𝜑⟩⟩.

(Contributed by Mario Carneiro, 14-Jul-2013.)

Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑚𝑚 ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚𝑚 ω) ∣ ∀𝑧𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚𝑚 ω) ∣ (𝑎𝑖)𝑒(𝑎𝑗)})}) ↾ suc ω))

Definitiondf-sate 30580* A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable 𝑛. (Contributed by Mario Carneiro, 14-Jul-2013.)
Sat = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))

Definitiondf-fmla 30581 Define the predicate which defines the set of valid Godel formulas. The parameter 𝑛 defines the maximum height of the formulas: the set (Fmla‘∅) is all formulas of the form 𝑥 = 𝑦 or 𝑥𝑦 (which in our coding scheme is the set ({∅, 1𝑜} × (ω × ω)); see df-sat 30579 for the full coding scheme), and each extra level adds to the complexity of the formulas in (Fmla‘𝑛). (Fmla‘ω) = 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛))

Syntaxcgon 30582 The Godel-set of negation. (Note that this is not a wff.)
class ¬𝑔𝑈

Syntaxcgoa 30583 The Godel-set of conjunction.
class 𝑔

Syntaxcgoi 30584 The Godel-set of implication.
class 𝑔

Syntaxcgoo 30585 The Godel-set of disjunction.
class 𝑔

Syntaxcgob 30586 The Godel-set of equivalence.
class 𝑔

Syntaxcgoq 30587 The Godel-set of equality.
class =𝑔

Syntaxcgox 30588 The Godel-set of existential quantification. (Note that this is not a wff.)
class 𝑔𝑁𝑈

Definitiondf-gonot 30589 Define the Godel-set of negation. Here the argument 𝑈 is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
¬𝑔𝑈 = (𝑈𝑔𝑈)

Definitiondf-goan 30590* Define the Godel-set of conjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ¬𝑔(𝑢𝑔𝑣))

Definitiondf-goim 30591* Define the Godel-set of implication. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢𝑔¬𝑔𝑣))

Definitiondf-goor 30592* Define the Godel-set of disjunction. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (¬𝑔𝑢𝑔 𝑣))

Definitiondf-gobi 30593* Define the Godel-set of equivalence. Here the arguments 𝑈 and 𝑉 are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢𝑔 𝑣)∧𝑔(𝑣𝑔 𝑢)))

Definitiondf-goeq 30594* Define the Godel-set of equality. Here the arguments 𝑥 = ⟨𝑁, 𝑃 correspond to vN and vP , so (∅=𝑔1𝑜) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2590 to introduce equality as a defined notion in terms of 𝑔. The expression suc (𝑢𝑣) = max (𝑢, 𝑣) + 1 here is a convenient way of getting a dummy variable distinct from 𝑢 and 𝑣. (Contributed by Mario Carneiro, 14-Jul-2013.)
=𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ suc (𝑢𝑣) / 𝑤𝑔𝑤((𝑤𝑔𝑢) ↔𝑔 (𝑤𝑔𝑣)))

Definitiondf-goex 30595 Define the Godel-set of existential quantification. Here 𝑁 ∈ ω corresponds to vN , and 𝑈 represents another formula, and this expression is [∃𝑥𝜑] = ∃𝑔𝑁𝑈 where 𝑥 is the 𝑁-th variable, 𝑈 = [𝜑] is the code for 𝜑. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
𝑔𝑁𝑈 = ¬𝑔𝑔𝑁¬𝑔𝑈

Definitiondf-prv 30596* Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀𝑚 ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by Mario Carneiro, 14-Jul-2013.)
⊧ = {⟨𝑚, 𝑢⟩ ∣ (𝑚 Sat 𝑢) = (𝑚𝑚 ω)}

21.5.11  Models of ZF

Syntaxcgze 30597 The Axiom of Extensionality.
class AxExt

Syntaxcgzr 30598 The Axiom Scheme of Replacement.
class AxRep

Syntaxcgzp 30599 The Axiom of Power Sets.
class AxPow

Syntaxcgzu 30600 The Axiom of Unions.
class AxUn

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 42301-42360
 Copyright terms: Public domain < Previous  Next >