P. 437 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43601-43700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clsneifv3 43601*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ 𝑋 ∈ (𝐾‘(𝐵 ∖ 𝑠))})
Theorem	clsneifv4 43602*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ 𝐷)    &   ⊢ (𝜑 → 𝐾𝐻𝑁)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)})
Theorem	neicvgbex 43603	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 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝐵 ∈ V)
Theorem	neicvgrcomplex 43604	The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
Theorem	neicvgf1o 43605*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝐻:(𝒫 𝒫 𝐵 ↑m 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgnvo 43606*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → ◡𝐻 = 𝐻)
Theorem	neicvgnvor 43607*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑀𝐻𝑁)
Theorem	neicvgmex 43608*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgnex 43609*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    ⇒   ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵))
Theorem	neicvgel1 43610*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → (𝑆 ∈ (𝑁‘𝑋) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑀‘𝑋)))
Theorem	neicvgel2 43611*	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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)    ⇒   ⊢ (𝜑 → ((𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋) ↔ ¬ 𝑆 ∈ (𝑀‘𝑋)))
Theorem	neicvgfv 43612*	The value of the neighborhoods (convergents) in terms of the convergents (neighborhoods) function. (Contributed by RP, 27-Jun-2021.)
⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))    &   ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))    &   ⊢ 𝐷 = (𝑃‘𝐵)    &   ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)    &   ⊢ 𝐺 = (𝐵𝑂𝒫 𝐵)    &   ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺))    &   ⊢ (𝜑 → 𝑁𝐻𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) = {𝑠 ∈ 𝒫 𝐵 ∣ ¬ (𝐵 ∖ 𝑠) ∈ (𝑀‘𝑋)})
Theorem	ntrrn 43613	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 𝐼 ⊆ 𝐽)
Theorem	ntrf 43614	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 𝑋⟶𝐽)
Theorem	ntrf2 43615	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)
Theorem	ntrelmap 43616	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 ↑m 𝒫 𝑋))
Theorem	clsf2 43617	The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 22965. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    ⇒   ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋)
Theorem	clselmap 43618	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 𝑋 ↑m 𝒫 𝑋))
Theorem	dssmapntrcls 43619*	The interior and closure operators on a topology are duals of each other. See also kur14lem2 34870. (Contributed by RP, 21-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝑋)    ⇒   ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))
Theorem	dssmapclsntr 43620*	The closure and interior operators on a topology are duals of each other. See also kur14lem2 34870. (Contributed by RP, 22-Apr-2021.)
⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (cls‘𝐽)    &   ⊢ 𝐼 = (int‘𝐽)    &   ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))    &   ⊢ 𝐷 = (𝑂‘𝑋)    ⇒   ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼))
21.34.7.3  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 22915 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems.
Theorem	gneispa 43621*	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 → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))
Theorem	gneispb 43622*	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 ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
Theorem	gneispace2 43623*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace3 43624*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ ((Fun 𝐹 ∧ ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
Theorem	gneispace 43625*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹‘𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))))
Theorem	gneispacef 43626*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacef2 43627*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
Theorem	gneispacefun 43628*	A generic neighborhood space is a function. (Contributed by RP, 15-Apr-2021.)
⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → Fun 𝐹)
Theorem	gneispacern 43629*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
Theorem	gneispacern2 43630*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
Theorem	gneispace0nelrn 43631*	A generic neighborhood space has a nonempty 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
Theorem	gneispace0nelrn2 43632*	A generic neighborhood space has a nonempty 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅)
Theorem	gneispace0nelrn3 43633*	A generic neighborhood space has a nonempty 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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ ran 𝐹)
Theorem	gneispaceel 43634*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛)
Theorem	gneispaceel2 43635*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁)
Theorem	gneispacess 43636*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))
Theorem	gneispacess2 43637*	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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}    ⇒   ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))
21.34.8  Exploring Higher Homotopy via Kerodon See https://kerodon.net/ for a work in progress by Jacob Lurie.
21.34.8.1  Simplicial Sets See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁.
Theorem	k0004lem1 43638	Application of ssin 4226 to range of a function. (Contributed by RP, 1-Apr-2021.)
⊢ (𝐷 = (𝐵 ∩ 𝐶) → ((𝐹:𝐴⟶𝐵 ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹:𝐴⟶𝐷))
Theorem	k0004lem2 43639	A mapping with a particular restricted range is also a mapping to that range. (Contributed by RP, 1-Apr-2021.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵) → ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ (𝐹 “ 𝐴) ⊆ 𝐶) ↔ 𝐹 ∈ (𝐶 ↑m 𝐴)))
Theorem	k0004lem3 43640	When the value of a mapping on a singleton is known, the mapping is a completely known singleton. (Contributed by RP, 2-Apr-2021.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵) → ((𝐹 ∈ (𝐵 ↑m {𝐴}) ∧ (𝐹‘𝐴) = 𝐶) ↔ 𝐹 = {⟨𝐴, 𝐶⟩}))
Theorem	k0004val 43641*	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))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) = {𝑡 ∈ ((0[,]1) ↑m (1...(𝑁 + 1))) ∣ Σ𝑘 ∈ (1...(𝑁 + 1))(𝑡‘𝑘) = 1})
Theorem	k0004ss1 43642*	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))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (ℝ ↑m (1...(𝑁 + 1))))
Theorem	k0004ss2 43643*	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))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(ℝ^‘(1...(𝑁 + 1)))))
Theorem	k0004ss3 43644*	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))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐴‘𝑁) ⊆ (Base‘(𝔼hil‘(𝑁 + 1))))
Theorem	k0004val0 43645*	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))(𝑡‘𝑘) = 1})    ⇒   ⊢ (𝐴‘0) = {{⟨1, 1⟩}}
21.35  Mathbox for Stanislas Polu
Theorem	inductionexd 43646	Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5))
21.35.1  IMO Problems
21.35.1.1  IMO 1972 B2
Theorem	wwlemuld 43647	Natural deduction form of lemul2d 13087. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝐶 · 𝐴) ≤ (𝐶 · 𝐵))    &   ⊢ (𝜑 → 0 < 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	leeq1d 43648	Specialization of breq1d 5154 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝐶)
Theorem	leeq2d 43649	Specialization of breq2d 5156 to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐷)
Theorem	absmulrposd 43650	Specialization of absmuld with absidd 15396. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = (𝐴 · (abs‘𝐵)))
Theorem	imadisjld 43651	Natural dduction form of one side of imadisj 6079. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)
Theorem	wnefimgd 43652	The image of a mapping from A is nonempty if A is nonempty. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅)
Theorem	fco2d 43653	Natural deduction form of fco2 6744. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	wfximgfd 43654	The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))
Theorem	extoimad 43655*	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.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ (abs “ (𝐹 “ ℝ))𝑥 ≤ 𝐶)
Theorem	imo72b2lem0 43656*	Lemma for imo72b2 43663. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ((𝐹‘(𝐴 + 𝐵)) + (𝐹‘(𝐴 − 𝐵))) = (2 · ((𝐹‘𝐴) · (𝐺‘𝐵))))    &   ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)    ⇒   ⊢ (𝜑 → ((abs‘(𝐹‘𝐴)) · (abs‘(𝐺‘𝐵))) ≤ sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
Theorem	suprleubrd 43657*	Natural deduction form of specialized suprleub 12205. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝑧 ≤ 𝐵)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ≤ 𝐵)
Theorem	imo72b2lem2 43658*	Lemma for imo72b2 43663. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑧 ∈ ℝ (abs‘(𝐹‘𝑧)) ≤ 𝐶)    ⇒   ⊢ (𝜑 → sup((abs “ (𝐹 “ ℝ)), ℝ, < ) ≤ 𝐶)
Theorem	suprlubrd 43659*	Natural deduction form of specialized suprlub 12203. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 𝐵 < 𝑧)    ⇒   ⊢ (𝜑 → 𝐵 < sup(𝐴, ℝ, < ))
Theorem	imo72b2lem1 43660*	Lemma for imo72b2 43663. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)    &   ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)    ⇒   ⊢ (𝜑 → 0 < sup((abs “ (𝐹 “ ℝ)), ℝ, < ))
Theorem	lemuldiv3d 43661	'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴))
Theorem	lemuldiv4d 43662	'Less than or equal to' relationship between division and multiplication. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐵 ≤ (𝐶 / 𝐴))    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) ≤ 𝐶)
Theorem	imo72b2 43663*	IMO 1972 B2. (14th International Mathematical Olympiad in Poland, problem B2). (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∀𝑢 ∈ ℝ ∀𝑣 ∈ ℝ ((𝐹‘(𝑢 + 𝑣)) + (𝐹‘(𝑢 − 𝑣))) = (2 · ((𝐹‘𝑢) · (𝐺‘𝑣))))    &   ⊢ (𝜑 → ∀𝑦 ∈ ℝ (abs‘(𝐹‘𝑦)) ≤ 1)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐹‘𝑥) ≠ 0)    ⇒   ⊢ (𝜑 → (abs‘(𝐺‘𝐵)) ≤ 1)
21.35.2  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 11242 1red 11240 readdcld 11268 remulcld 11269 eqcomd 2731.
Theorem	int-addcomd 43664	AdditionCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐶) = (𝐶 + 𝐴))
Theorem	int-addassocd 43665	AdditionAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + 𝐷))
Theorem	int-addsimpd 43666	AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 0 = (𝐴 − 𝐵))
Theorem	int-mulcomd 43667	MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))
Theorem	int-mulassocd 43668	MultiplicationAssociativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 · (𝐶 · 𝐷)) = ((𝐴 · 𝐶) · 𝐷))
Theorem	int-mulsimpd 43669	MultiplicationSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → 1 = (𝐴 / 𝐵))
Theorem	int-leftdistd 43670	AdditionMultiplicationLeftDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ((𝐶 + 𝐷) · 𝐵) = ((𝐶 · 𝐴) + (𝐷 · 𝐴)))
Theorem	int-rightdistd 43671	AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐵 · (𝐶 + 𝐷)) = ((𝐴 · 𝐶) + (𝐴 · 𝐷)))
Theorem	int-sqdefd 43672	SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2))
Theorem	int-mul11d 43673	First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 1) = 𝐵)
Theorem	int-mul12d 43674	Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (1 · 𝐴) = 𝐵)
Theorem	int-add01d 43675	First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 + 0) = 𝐵)
Theorem	int-add02d 43676	Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (0 + 𝐴) = 𝐵)
Theorem	int-sqgeq0d 43677	SquareGEQZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ (𝐴 · 𝐵))
Theorem	int-eqprincd 43678	PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷))
Theorem	int-eqtransd 43679	EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	int-eqmvtd 43680	EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷))    ⇒   ⊢ (𝜑 → 𝐶 = (𝐵 − 𝐷))
Theorem	int-eqineqd 43681	EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≤ 𝐴)
Theorem	int-ineqmvtd 43682	IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷))    ⇒   ⊢ (𝜑 → (𝐵 − 𝐷) ≤ 𝐶)
Theorem	int-ineq1stprincd 43683	FirstPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐷 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐷) ≤ (𝐴 + 𝐶))
Theorem	int-ineq2ndprincd 43684	SecondPrincipleOfInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐶) ≤ (𝐴 · 𝐶))
Theorem	int-ineqtransd 43685	InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝐴)
21.35.3  N-Digit Addition Proof Generator This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12717 addcomli 11431 00id 11414 addridi 11426 addlidi 11427 eqid 2725 dec0h 12724 decadd 12756 decaddc 12757.
Theorem	unitadd 43686	Theorem used in conjunction with decaddc 12757 to absorb carry when generating n-digit addition synthetic proofs. (Contributed by Stanislas Polu, 7-Apr-2020.)
⊢ (𝐴 + 𝐵) = 𝐹    &   ⊢ (𝐶 + 1) = 𝐵    &   ⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    ⇒   ⊢ ((𝐴 + 𝐶) + 1) = 𝐹
21.35.4  AM-GM (for k = 2,3,4)
Theorem	gsumws3 43687	Valuation of a length 3 word in a monoid. (Contributed by Stanislas Polu, 9-Sep-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ 𝑈 ∈ 𝐵))) → (𝐺 Σg ⟨“𝑆𝑇𝑈”⟩) = (𝑆 + (𝑇 + 𝑈)))
Theorem	gsumws4 43688	Valuation of a length 4 word in a monoid. (Contributed by Stanislas Polu, 10-Sep-2020.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ Mnd ∧ (𝑆 ∈ 𝐵 ∧ (𝑇 ∈ 𝐵 ∧ (𝑈 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)))) → (𝐺 Σg ⟨“𝑆𝑇𝑈𝑉”⟩) = (𝑆 + (𝑇 + (𝑈 + 𝑉))))
Theorem	amgm2d 43689	Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 26935. (Contributed by Stanislas Polu, 8-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2))
Theorem	amgm3d 43690	Arithmetic-geometric mean inequality for 𝑛 = 3. (Contributed by Stanislas Polu, 11-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · (𝐵 · 𝐶))↑𝑐(1 / 3)) ≤ ((𝐴 + (𝐵 + 𝐶)) / 3))
Theorem	amgm4d 43691	Arithmetic-geometric mean inequality for 𝑛 = 4. (Contributed by Stanislas Polu, 11-Sep-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((𝐴 · (𝐵 · (𝐶 · 𝐷)))↑𝑐(1 / 4)) ≤ ((𝐴 + (𝐵 + (𝐶 + 𝐷))) / 4))
21.36  Mathbox for Rohan Ridenour
21.36.1  Misc
Theorem	spALT 43692	sp 2171 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2171 instead. (New usage is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	elnelneqd 43693	Two classes are not equal if there is an element of one which is not an element of the other. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
Theorem	elnelneq2d 43694	Two classes are not equal if one but not the other is an element of a given class. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
Theorem	rr-spce 43695*	Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.)
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	rexlimdvaacbv 43696*	Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3146. (Contributed by Rohan Ridenour, 3-Aug-2023.)
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimddvcbvw 43697*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 43696. The equivalent of this theorem without the bound variable change is rexlimddv 3151. Version of rexlimddvcbv 43698 with a disjoint variable condition, which does not require ax-13 2365. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by Gino Giotto, 2-Apr-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	rexlimddvcbv 43698*	Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 43696. The equivalent of this theorem without the bound variable change is rexlimddv 3151. Usage of this theorem is discouraged because it depends on ax-13 2365, see rexlimddvcbvw 43697 for a weaker version that does not require it. (Contributed by Rohan Ridenour, 3-Aug-2023.) (New usage is discouraged.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	rr-elrnmpt3d 43699*	Elementhood in an image set. (Contributed by Rohan Ridenour, 11-Aug-2023.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 ∈ ran 𝐹)
Theorem	finnzfsuppd 43700*	If a function is zero outside of a finite set, it has finite support. (Contributed by Rohan Ridenour, 13-May-2024.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑥 ∈ 𝐴 ∨ (𝐹‘𝑥) = 𝑍))    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)