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Type | Label | Description |
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Statement | ||
Theorem | iinllyconn 32501 | The unit interval is locally connected. (Contributed by Mario Carneiro, 6-Jul-2015.) |
⊢ II ∈ 𝑛-Locally Conn | ||
Syntax | ccvm 32502 | Extend class notation with the class of covering maps. |
class CovMap | ||
Definition | df-cvm 32503* | 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 32504 | Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ CovMap Fn (Top × Top) | ||
Theorem | cvmscbv 32505* | 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 32506* | 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 32507 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top) | ||
Theorem | cvmtop2 32508 | Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐽 ∈ Top) | ||
Theorem | cvmcn 32509 | A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽)) | ||
Theorem | cvmcov 32510* | 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 32511* | 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 32512* | One direction of cvmsval 32513. (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 32513* | 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 32514* | 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 32515* | 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 32516* | 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 32517* | 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 32518* | 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 32519* | 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 32520* | 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 32521* | 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 32522* | 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 32523* | 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 32524* | 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 32525* | Lemma for cvmopn 32527. (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 32526* | Lemma for cvmfo 32547. (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 32527 | 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 32528* | Lemma for cvmliftmo 32531. (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 32529* | Lemma for cvmliftmo 32531. (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 32530 | 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 32531* | 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 32532* | Lemma for cvmlift 32546. In cvmliftlem15 32545, 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 32533* | Lemma for cvmlift 32546. 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 32534* | Lemma for cvmlift 32546. 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 32535* | Lemma for cvmlift 32546. 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 32536* | Lemma for cvmlift 32546. 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 32537* | Lemma for cvmlift 32546. Induction step for cvmliftlem7 32538. 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 32538* | Lemma for cvmlift 32546. Prove by induction that every 𝑄 function is well-defined (we can immediately follow this theorem with cvmliftlem6 32537 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 32539* | Lemma for cvmlift 32546. 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 32540* | Lemma for cvmlift 32546. 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 32541* | Lemma for cvmlift 32546. 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 32537, cvmliftlem7 32538 (to show it is a function and a lift), cvmliftlem8 32539 (to show it is continuous), and cvmliftlem9 32540 (to show that different 𝑄 functions agree on the intersection of their domains, so that the pasting lemma paste 21902 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 32542* | Lemma for cvmlift 32546. (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 32543* | Lemma for cvmlift 32546. 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 32544* | Lemma for cvmlift 32546. Putting the results of cvmliftlem11 32542, cvmliftlem13 32543 and cvmliftmo 32531 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 32545* | Lemma for cvmlift 32546. Discharge the assumptions of cvmliftlem14 32544. 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 23570, 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 8762 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 32544. (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 32546* | 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 32532 thru cvmliftlem15 32545. (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 32547 | 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 32548* | 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 32549* | Lemma for cvmlift2 32563. (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 32550* | Lemma for cvmlift2 32563 and cvmlift3 32575. (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 32551* | Lemma for cvmlift2 32563. (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 32552* | Lemma for cvmlift2 32563. (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 32553* | Lemma for cvmlift2 32563. (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 32554* | Lemma for cvmlift2 32563. (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 32555* | Lemma for cvmlift2 32563. (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 32556* | Lemma for cvmlift2 32563. (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 32557* | Lemma for cvmlift2 32563. (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 32558* | Lemma for cvmlift2 32563. (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 32559* | Lemma for cvmlift2 32563. (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 32560* | Lemma for cvmlift2 32563. (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 32561* | Lemma for cvmlift2 32563. (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 32562* | Lemma for cvmlift2 32563. (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 32563* | A two-dimensional version of cvmlift 32546. 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 32564* | Lemma for cvmliftpht 32565. (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 32565* | 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 32566* | Lemma for cvmlift3 32575. (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 32567* | Lemma for cvmlift2 32563. (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 32568* | Lemma for cvmlift2 32563. (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 32569* | Lemma for cvmlift2 32563. (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 32570* | Lemma for cvmlift2 32563. (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 32571* | Lemma for cvmlift3 32575. (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 32572* | Lemma for cvmlift3 32575. (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 32573* | Lemma for cvmlift2 32563. (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 32574* | Lemma for cvmlift2 32563. (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 32575* | A general version of cvmlift 32546. 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)) | ||
Theorem | snmlff 32576* | The function 𝐹 from snmlval 32578 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 32577* | The function 𝐹 from snmlval 32578 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 32578* | 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 32579* | 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)) | ||
Syntax | cgoe 32580 | The Godel-set of membership. |
class ∈𝑔 | ||
Syntax | cgna 32581 | The Godel-set for the Sheffer stroke. |
class ⊼𝑔 | ||
Syntax | cgol 32582 | The Godel-set of universal quantification. (Note that this is not a wff.) |
class ∀𝑔𝑁𝑈 | ||
Syntax | csat 32583 | The satisfaction function. |
class Sat | ||
Syntax | cfmla 32584 | The formula set predicate. |
class Fmla | ||
Syntax | csate 32585 | The ∈-satisfaction function. |
class Sat∈ | ||
Syntax | cprv 32586 | The "proves" relation. |
class ⊧ | ||
Definition | df-goel 32587 | 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〉) | ||
Definition | df-gona 32588 | Define the Godel-set for the Sheffer stroke NAND. Here the arguments 𝑥 = 〈𝑈, 𝑉〉 are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦ 〈1o, 𝑥〉) | ||
Definition | df-goal 32589 | 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〉〉 | ||
Definition | df-sat 32590* |
Define the satisfaction predicate. This recursive construction builds up
a function over wff codes (see satff 32657) 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
(Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ ∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω)) | ||
Definition | df-sate 32591* | 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)) | ||
Definition | df-fmla 32592 | 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 (which in our coding scheme is the set ({∅} × (ω × ω)); see df-sat 32590 for the full coding scheme), see fmla0 32629, and each extra level adds to the complexity of the formulas in (Fmla‘𝑛), see fmlasuc 32633. Remark: it is sufficient to have atomic formulas of the form 𝑥 ∈ 𝑦 only, because equations (formulas of the form 𝑥 = 𝑦), which are required as (atomic) formulas, can be introduced as a defined notion in terms of ∈𝑔, see df-goeq 32691. (Fmla‘ω) = ∪ 𝑛 ∈ ω(Fmla‘𝑛) is the set of all valid formulas, see fmla 32628. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat ∅)‘𝑛)) | ||
Definition | df-prv 32593* | Define the "proves" relation on a set. A wff is true in a model 𝑀 if for every valuation 𝑠 ∈ (𝑀 ↑m ω), the interpretation of the wff using the membership relation on 𝑀 is true. Since ⊧ is defined in terms of the interpretations making the given formula true, it is not defined on the empty "model" 𝑀 = ∅, since there are no interpretations. In particular, the empty set on the LHS of ⊧ should not be interpreted as the empty model. Statement prv0 32677 shows that our definition yields ∅⊧𝑈 for all formulas, though of course the formula ∃𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ ⊧ = {〈𝑚, 𝑢〉 ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑m ω)} | ||
Theorem | goel 32594 | A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi ∈ vj is coded as 〈∅, 〈𝑖, 𝑗〉〉. (Contributed by AV, 15-Sep-2023.) |
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) = 〈∅, 〈𝐼, 𝐽〉〉) | ||
Theorem | goelel3xp 32595 | A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.) |
⊢ ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼∈𝑔𝐽) ∈ (ω × (ω × ω))) | ||
Theorem | goeleq12bg 32596 | Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.) |
⊢ (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼∈𝑔𝐽) = (𝑀∈𝑔𝑁) ↔ (𝐼 = 𝑀 ∧ 𝐽 = 𝑁))) | ||
Theorem | gonafv 32597 | The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴⊼𝑔𝐵) = 〈1o, 〈𝐴, 𝐵〉〉) | ||
Theorem | goaleq12d 32598 | Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.) |
⊢ (𝜑 → 𝑀 = 𝑁) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵) | ||
Theorem | gonanegoal 32599 | The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.) |
⊢ (𝑎⊼𝑔𝑏) ≠ ∀𝑔𝑖𝑢 | ||
Theorem | satf 32600* | The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 14-Sep-2023.) |
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)) |
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