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Theorem List for Metamath Proof Explorer - 38701-38800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremntrneiel2 38701* Membership in iterated interior of a set is equivalent to there existing a particular neighborhood of that member such that points are members of that neighborhood if and only if the set is a neighborhood of each of those points. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐼‘(𝐼𝑆)) ↔ ∃𝑢 ∈ (𝑁𝑋)∀𝑦𝐵 (𝑦𝑢𝑆 ∈ (𝑁𝑦))))

Theoremntrneineine0 38702* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ ∅))

Theoremntrneineine1 38703* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at not all subsets are (pseudo-)neighborboods hold equally. (Contributed by RP, 1-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑥𝐵𝑠 ∈ 𝒫 𝐵 ¬ 𝑥 ∈ (𝐼𝑠) ↔ ∀𝑥𝐵 (𝑁𝑥) ≠ 𝒫 𝐵))

Theoremntrneicls00 38704* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼𝐵) = 𝐵 ↔ ∀𝑥𝐵 𝐵 ∈ (𝑁𝑥)))

Theoremntrneicls11 38705* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior of the empty set is the empty set hold equally. (Contributed by RP, 2-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → ((𝐼‘∅) = ∅ ↔ ∀𝑥𝐵 ¬ ∅ ∈ (𝑁𝑥)))

Theoremntrneiiso 38706* If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that the interior function is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝑠𝑡 → (𝐼𝑠) ⊆ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑠𝑡) → 𝑡 ∈ (𝑁𝑥))))

Theoremntrneik2 38707* An interior function is contracting if and only if all the neighborhoods of a point contain that point. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼𝑠) ⊆ 𝑠 ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) → 𝑥𝑠)))

Theoremntrneix2 38708* An interior (closure) function is expansive if and only if all subsets which contain a point are neighborhoods (convergents) of that point. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑥𝑠𝑠 ∈ (𝑁𝑥))))

Theoremntrneikb 38709* The interiors of disjoint sets are disjoint if and only if the neighborhoods of every point contain no disjoint sets. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = ∅ → ((𝐼𝑠) ∩ (𝐼𝑡)) = ∅) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ≠ ∅)))

Theoremntrneixb 38710* The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → ((𝐼𝑠) ∪ (𝐼𝑡)) = 𝐵) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) = 𝐵 → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik3 38711* The intersection of interiors of any pair is a subset of the interior of the intersection if and only if the intersection of any two neighborhoods of a point is also a neighborhood. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝐼𝑠) ∩ (𝐼𝑡)) ⊆ (𝐼‘(𝑠𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)) → (𝑠𝑡) ∈ (𝑁𝑥))))

Theoremntrneix3 38712* The closure of the union of any pair is a subset of the union of closures if and only if the union of any pair belonging to the convergents of a point implies at least one of the pair belongs to the the convergents of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) ⊆ ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) → (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik13 38713* The interior of the intersection of any pair equals intersection of interiors if and only if the intersection of any pair belonging to the neighborhood of a point is equivalent to both of the pair belonging to the neighborhood of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∩ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∧ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneix13 38714* The closure of the union of any pair is equal to the union of closures if and only if the union of any pair belonging to the convergents of a point if equivalent to at least one of the pain belonging to the convergents of that point. (Contributed by RP, 19-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵(𝐼‘(𝑠𝑡)) = ((𝐼𝑠) ∪ (𝐼𝑡)) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵𝑡 ∈ 𝒫 𝐵((𝑠𝑡) ∈ (𝑁𝑥) ↔ (𝑠 ∈ (𝑁𝑥) ∨ 𝑡 ∈ (𝑁𝑥)))))

Theoremntrneik4w 38715* Idempotence of the interior function is equivalent to saying a set is a neighborhood of a point if and only if the interior of the set is a neighborhood of a point. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ (𝐼𝑠) ∈ (𝑁𝑥))))

Theoremntrneik4 38716* Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   (𝜑𝐼𝐹𝑁)       (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼𝑠)) = (𝐼𝑠) ↔ ∀𝑥𝐵𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁𝑥) ↔ ∃𝑢 ∈ (𝑁𝑥)∀𝑦𝐵 (𝑦𝑢𝑠 ∈ (𝑁𝑦)))))

Theoremclsneibex 38717 If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐵 ∈ V)

Theoremclsneircomplex 38718 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremclsneif1o 38719* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the operator is a one-to-one, onto mapping. (Contributed by RP, 5-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))

Theoremclsneicnv 38720* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then the converse of the operator is known. (Contributed by RP, 5-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐻 = (𝐷 ∘ (𝐵𝑂𝒫 𝐵)))

Theoremclsneikex 38721* If closure and neighborhoods functions are related, the closure function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝐾 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))

Theoremclsneinex 38722* If closure and neighborhoods functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremclsneiel1 38723* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾𝑆) ↔ ¬ (𝐵𝑆) ∈ (𝑁𝑋)))

Theoremclsneiel2 38724* If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of the complement of a subset is equivalent to the subset not being a neighborhood of the point. (Contributed by RP, 7-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑋 ∈ (𝐾‘(𝐵𝑆)) ↔ ¬ 𝑆 ∈ (𝑁𝑋)))

Theoremclsneifv3 38725* Value of the neighborhoods (convergents) in terms of the closure (interior) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵𝑠))})

Theoremclsneifv4 38726* Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐻 = (𝐹𝐷)    &   (𝜑𝐾𝐻𝑁)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝐾𝑆) = {𝑥𝐵 ∣ ¬ (𝐵𝑆) ∈ (𝑁𝑥)})

Theoremneicvgbex 38727 If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐵 ∈ V)

Theoremneicvgrcomplex 38728 The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
𝐷 = (𝑃𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Theoremneicvgf1o 38729* If neighborhood and convergent functions are related by operator 𝐻, it is a one-to-one onto relation. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻:(𝒫 𝒫 𝐵𝑚 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgnvo 38730* If neighborhood and convergent functions are related by operator 𝐻, it is its own converse function. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝐻 = 𝐻)

Theoremneicvgnvor 38731* If neighborhood and convergent functions are related by operator 𝐻, the relationship holds with the functions swapped. (Contributed by RP, 11-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀𝐻𝑁)

Theoremneicvgmex 38732* If the neighborhoods and convergents functions are related, the convergents function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑀 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgnex 38733* If the neighborhoods and convergents functions are related, the neighborhoods function exists. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)       (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))

Theoremneicvgel1 38734* A subset being an element of a neighborhood of a point is equivalent to the complement of that subset not being a element of the convergent of that point. (Contributed by RP, 12-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → (𝑆 ∈ (𝑁𝑋) ↔ ¬ (𝐵𝑆) ∈ (𝑀𝑋)))

Theoremneicvgel2 38735* The complement of a subset being an element of a neighborhood at a point is equivalent to that subset not being a element of the convergent at that point. (Contributed by RP, 12-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)    &   (𝜑𝑆 ∈ 𝒫 𝐵)       (𝜑 → ((𝐵𝑆) ∈ (𝑁𝑋) ↔ ¬ 𝑆 ∈ (𝑀𝑋)))

Theoremneicvgfv 38736* The value of the neighborhoods (convergents) in terms of the the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗𝑚 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))    &   𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛𝑚 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛𝑜))))))    &   𝐷 = (𝑃𝐵)    &   𝐹 = (𝒫 𝐵𝑂𝐵)    &   𝐺 = (𝐵𝑂𝒫 𝐵)    &   𝐻 = (𝐹 ∘ (𝐷𝐺))    &   (𝜑𝑁𝐻𝑀)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵𝑠) ∈ (𝑀𝑋)})

Theoremntrrn 38737 The range of the interior function of a topology a subset of the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → ran 𝐼𝐽)

Theoremntrf 38738 The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)

Theoremntrf2 38739 The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼:𝒫 𝑋⟶𝒫 𝑋)

Theoremntrelmap 38740 The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐼 = (int‘𝐽)       (𝐽 ∈ Top → 𝐼 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))

Theoremclsf2 38741 The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 20900. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)       (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)

Theoremclselmap 38742 The closure function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)       (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))

Theoremdssmapntrcls 38743* The interior and closure operators on a topology are duals of each other. See also kur14lem2 31315. (Contributed by RP, 21-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))

Theoremdssmapclsntr 38744* The closure and interior operators on a topology are duals of each other. See also kur14lem2 31315. (Contributed by RP, 22-Apr-2021.)
𝑋 = 𝐽    &   𝐾 = (cls‘𝐽)    &   𝐼 = (int‘𝐽)    &   𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))    &   𝐷 = (𝑂𝑋)       (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))

20.27.4.3  Generic Neighborhood Spaces

Any neighborhood space is an open set topology and any open set topology is a neighborhood space. Seifert And Threlfall define a generic neighborhood space which is a superset of what is now generally used and related concepts and the following will show that those definitions apply to elements of Top.

Seifert And Threlfall do not allow neighborhood spaces on the empty set while sn0top 20851 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems.

Theoremgneispa 38745* Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert And Threlfall. (Contributed by RP, 5-Apr-2021.)
𝑋 = 𝐽       (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))

Theoremgneispb 38746* Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert And Threlfall. (Contributed by RP, 5-Apr-2021.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃})))

Theoremgneispace2 38747* The predicate that 𝐹 is a (generic) Seifert And Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))

Theoremgneispace3 38748* The predicate that 𝐹 is a (generic) Seifert And Threlfall neighborhood space. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))

Theoremgneispace 38749* The predicate that 𝐹 is a (generic) Seifert And Threlfall neighborhood space. (Contributed by RP, 14-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝑉 → (𝐹𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))))

Theoremgneispacef 38750* A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))

Theoremgneispacef2 38751* A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)

Theoremgneispacefun 38752* A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → Fun 𝐹)

Theoremgneispacern 38753* A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))

Theoremgneispacern2 38754* A generic neighborhood space has a range that is a subset of the powerset of the powerset of its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)

Theoremgneispace0nelrn 38755* A generic neighborhood space has a non-empty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅)

Theoremgneispace0nelrn2 38756* A generic neighborhood space has a non-empty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       ((𝐹𝐴𝑃 ∈ dom 𝐹) → (𝐹𝑃) ≠ ∅)

Theoremgneispace0nelrn3 38757* A generic neighborhood space has a non-empty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ¬ ∅ ∈ ran 𝐹)

Theoremgneispaceel 38758* Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)𝑝𝑛)

Theoremgneispaceel2 38759* Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       ((𝐹𝐴𝑃 ∈ dom 𝐹𝑁 ∈ (𝐹𝑃)) → 𝑃𝑁)

Theoremgneispacess 38760* All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))

Theoremgneispacess2 38761* All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}       (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))

20.27.5  Exploring Higher Homotopy via Kerodon

See https://kerodon.net/ for a work in progress by Jacob Lurie.

20.27.5.1  Simplicial Sets

See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁.

Theoremk0004lem1 38762 Application of ssin 3868 to range of a function. (Contributed by RP, 1-Apr-2021.)
(𝐷 = (𝐵𝐶) → ((𝐹:𝐴𝐵 ∧ (𝐹𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴𝐷))

Theoremk0004lem2 38763 A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.)
((𝐴𝑈𝐵𝑉𝐶𝐵) → ((𝐹 ∈ (𝐵𝑚 𝐴) ∧ (𝐹𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶𝑚 𝐴)))

Theoremk0004lem3 38764 When the value of a mapping on a singleton is known, the mapping is a completely known singleton. (Contributed by RP, 2-Apr-2021.)
((𝐴𝑈𝐵𝑉𝐶𝐵) → ((𝐹 ∈ (𝐵𝑚 {𝐴}) ∧ (𝐹𝐴) = 𝐶) ↔ 𝐹 = {⟨𝐴, 𝐶⟩}))

Theoremk0004val 38765* The topological simplex of dimension 𝑁 is the set of real vectors where the components are nonnegative and sum to 1. (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) = {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡𝑘) = 1})

Theoremk0004ss1 38766* The topological simplex of dimension 𝑁 is a subset of the real vectors of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (ℝ ↑𝑚 (1...(𝑁 + 1))))

Theoremk0004ss2 38767* The topological simplex of dimension 𝑁 is a subset of the base set of a real vector space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1)))))

Theoremk0004ss3 38768* The topological simplex of dimension 𝑁 is a subset of the base set of Euclidean space of dimension (𝑁 + 1). (Contributed by RP, 29-Mar-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝑁 ∈ ℕ0 → (𝐴𝑁) ⊆ (Base‘(𝔼hil‘(𝑁 + 1))))

Theoremk0004val0 38769* The topological simplex of dimension 0 is a singleton. (Contributed by RP, 2-Apr-2021.)
𝐴 = (𝑛 ∈ ℕ0 ↦ {𝑡 ∈ ((0[,]1) ↑𝑚 (1...(𝑛 + 1))) ∣ Σ𝑘 ∈ (1...(𝑛 + 1))(𝑡𝑘) = 1})       (𝐴‘0) = {{⟨1, 1⟩}}

20.28  Mathbox for Stanislas Polu

Theoreminductionexd 38770 Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))

20.28.1  IMO Problems

20.28.1.1  IMO 1972 B2

Theoremwwlemuld 38771 Natural deduction form of lemul2d 11954. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))    &   (𝜑 → 0 < 𝐶)       (𝜑𝐴𝐵)

Theoremleeq1d 38772 Specialization of breq1d 4695 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐵𝐶)

Theoremleeq2d 38773 Specialization of breq2d 4697 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴𝐶)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐴𝐷)

Theoremabsmulrposd 38774 Specialization of absmuld with absidd 14205. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (abs‘(𝐴 · 𝐵)) = (𝐴 · (abs‘𝐵)))

Theoremimadisjld 38775 Natural dduction form of one side of imadisj 5519. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (dom 𝐴𝐵) = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremimadisjlnd 38776 Natural deduction form of one negated side of imadisj 5519. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (dom 𝐴𝐵) ≠ ∅)       (𝜑 → (𝐴𝐵) ≠ ∅)

Theoremwnefimgd 38777 The image of a mapping from A is non empty if A is non empty. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ≠ ∅)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐴) ≠ ∅)

Theoremfco2d 38778 Natural deduction form of fco2 6097. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐺:𝐴𝐵)    &   (𝜑 → (𝐹𝐵):𝐵𝐶)       (𝜑 → (𝐹𝐺):𝐴𝐶)

Theoremfvco3d 38779 Natural deduction form of fvco3 6314. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐺:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))

Theoremwfximgfd 38780 The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐶𝐴)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝐹𝐶) ∈ (𝐹𝐴))

Theoremextoimad 38781* If |f(x)| <= C for all x then it applies to all x in the image of |f(x)| (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 𝐶)       (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥𝐶)

Theoremimo72b2lem0 38782* Lemma for imo72b2 38792. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐺:ℝ⟶ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴𝐵))) = (2 · ((𝐹𝐴) · (𝐺𝐵))))    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)       (𝜑 → ((abs‘(𝐹𝐴)) · (abs‘(𝐺𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))

Theoremsuprleubrd 38783* Natural deduction form of specialized suprleub 11027. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∀𝑧𝐴 𝑧𝐵)       (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵)

Theoremimo72b2lem2 38784* Lemma for imo72b2 38792. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹𝑧)) ≤ 𝐶)       (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶)

Theoremsyldbl2 38785 Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)
((𝜑𝜓) → (𝜓𝜃))       ((𝜑𝜓) → 𝜃)

Theoremfunfvima2d 38786 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))

Theoremsuprlubrd 38787* Natural deduction form of specialized suprlub 11025. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑦𝑥)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑧𝐴 𝐵 < 𝑧)       (𝜑𝐵 < sup(𝐴, ℝ, < ))

Theoremimo72b2lem1 38788* Lemma for imo72b2 38792. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)       (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))

Theoremlemuldiv3d 38789 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑 → (𝐵 · 𝐴) ≤ 𝐶)    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑𝐵 ≤ (𝐶 / 𝐴))

Theoremlemuldiv4d 38790 'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐵 ≤ (𝐶 / 𝐴))    &   (𝜑 → 0 < 𝐴)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → (𝐵 · 𝐴) ≤ 𝐶)

Theoremrspcdvinvd 38791* If something is true for all then it's true for some class. (Contributed by Stanislas Polu, 9-Mar-2020.)
((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   (𝜑𝐴𝐵)    &   (𝜑 → ∀𝑥𝐵 𝜓)       (𝜑𝜒)

Theoremimo72b2 38792* IMO 1972 B2. (14th International Mathemahics Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐺:ℝ⟶ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢𝑣))) = (2 · ((𝐹𝑢) · (𝐺𝑣))))    &   (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹𝑦)) ≤ 1)    &   (𝜑 → ∃𝑥 ∈ ℝ (𝐹𝑥) ≠ 0)       (𝜑 → (abs‘(𝐺𝐵)) ≤ 1)

20.28.2  INT Inequalities Proof Generator

This section formalizes theorems necessary to reproduce the equality and inequality generator described in "Neural Theorem Proving on Inequality Problems" http://aitp-conference.org/2020/abstract/paper_18.pdf.

Other theorems required: 0red 10079 1red 10093 readdcld 10107 remulcld 10108 eqcomd 2657.

(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐴))

(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))

(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → 0 = (𝐴𝐵))

Theoremint-mulcomd 38796 MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))

Theoremint-mulassocd 38797 MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))

Theoremint-mulsimpd 38798 MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)    &   (𝜑𝐵 ≠ 0)       (𝜑 → 1 = (𝐴 / 𝐵))

Theoremint-leftdistd 38799 AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴)))

Theoremint-rightdistd 38800 AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷)))

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