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Type | Label | Description |
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Statement | ||
Theorem | neicvgbex 44101 | 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 44102 | 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 44103* | 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 44104* | 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 44105* | 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 44106* | 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 44107* | 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 44108* | 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 44109* | 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 44110* | 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 44111 | 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 44112 | 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 44113 | 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 44114 | 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 44115 | The closure function is a map from the powerset of the base set to itself. This is less precise than clsf 23071. (Contributed by RP, 22-Apr-2021.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (cls‘𝐽) ⇒ ⊢ (𝐽 ∈ Top → 𝐾:𝒫 𝑋⟶𝒫 𝑋) | ||
Theorem | clselmap 44116 | 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 44117* | The interior and closure operators on a topology are duals of each other. See also kur14lem2 35191. (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 44118* | The closure and interior operators on a topology are duals of each other. See also kur14lem2 35191. (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)) | ||
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 23021 is an example of a topology with an empty base set. This divergence is unlikely to pose serious problems. | ||
Theorem | gneispa 44119* | 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 44120* | 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 44121* | 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 44122* | 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 44123* | 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 44124* | 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 44125* | 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 44126* | 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 44127* | 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 44128* | 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 44129* | 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 44130* | 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 44131* | 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 44132* | 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 44133* | 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 44134* | 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 44135* | 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)) | ||
See https://kerodon.net/ for a work in progress by Jacob Lurie. | ||
See https://kerodon.net/tag/0004 for introduction to the topological simplex of dimension 𝑁. | ||
Theorem | k0004lem1 44136 | Application of ssin 4246 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 44137 | 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 44138 | 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 44139* | 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 44140* | 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 44141* | 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 44142* | 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 44143* | 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〉}} | ||
Theorem | inductionexd 44144 | Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝑁 ∈ ℕ → 3 ∥ ((4↑𝑁) + 5)) | ||
Theorem | wwlemuld 44145 | Natural deduction form of lemul2d 13118. (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 44146 | Specialization of breq1d 5157 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 44147 | Specialization of breq2d 5159 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 44148 | Specialization of absmuld with absidd 15457. (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 44149 | Natural dduction form of one side of imadisj 6099. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅) | ||
Theorem | wnefimgd 44150 | 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 44151 | Natural deduction form of fco2 6762. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶) | ||
Theorem | wfximgfd 44152 | 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 44153* | 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 44154* | Lemma for imo72b2 44161. (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 44155* | Natural deduction form of specialized suprleub 12231. (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 44156* | Lemma for imo72b2 44161. (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 44157* | Natural deduction form of specialized suprlub 12229. (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 44158* | Lemma for imo72b2 44161. (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 44159 | '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 44160 | '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 44161* | 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) | ||
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 11261 1red 11259 readdcld 11287 remulcld 11288 eqcomd 2740. | ||
Theorem | int-addcomd 44162 | 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 44163 | 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 44164 | AdditionSimplification generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 0 = (𝐴 − 𝐵)) | ||
Theorem | int-mulcomd 44165 | 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 44166 | 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 44167 | 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 44168 | 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 44169 | 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 44170 | SquareDefinition generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 𝐵) = (𝐴↑2)) | ||
Theorem | int-mul11d 44171 | First MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 · 1) = 𝐵) | ||
Theorem | int-mul12d 44172 | Second MultiplicationOne generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (1 · 𝐴) = 𝐵) | ||
Theorem | int-add01d 44173 | First AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 + 0) = 𝐵) | ||
Theorem | int-add02d 44174 | Second AdditionZero generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (0 + 𝐴) = 𝐵) | ||
Theorem | int-sqgeq0d 44175 | 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 44176 | PrincipleOfEquality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 + 𝐶) = (𝐵 + 𝐷)) | ||
Theorem | int-eqtransd 44177 | EqualityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → 𝐴 = 𝐶) | ||
Theorem | int-eqmvtd 44178 | 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 44179 | EquivalenceImpliesDoubleInequality generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≤ 𝐴) | ||
Theorem | int-ineqmvtd 44180 | 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 44181 | 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 44182 | 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 44183 | 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) | ||
This section formalizes theorems used in an n-digit addition proof generator. Other theorems required: deccl 12745 addcomli 11450 00id 11433 addridi 11445 addlidi 11446 eqid 2734 dec0h 12752 decadd 12784 decaddc 12785. | ||
Theorem | unitadd 44184 | Theorem used in conjunction with decaddc 12785 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 | ||
Theorem | gsumws3 44185 | 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 44186 | 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 44187 | Arithmetic-geometric mean inequality for 𝑛 = 2, derived from amgmlem 27047. (Contributed by Stanislas Polu, 8-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑐(1 / 2)) ≤ ((𝐴 + 𝐵) / 2)) | ||
Theorem | amgm3d 44188 | 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 44189 | 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)) | ||
Theorem | spALT 44190 | sp 2180 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2180 instead. (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | elnelneqd 44191 | 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 44192 | 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 44193* | Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | rexlimdvaacbv 44194* | Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3153. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
Theorem | rexlimddvcbvw 44195* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44194. The equivalent of this theorem without the bound variable change is rexlimddv 3158. Version of rexlimddvcbv 44196 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Revised by GG, 2-Apr-2024.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜃) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜃 ↔ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | rexlimddvcbv 44196* | Unpack a restricted existential assumption while changing the variable with implicit substitution. Similar to rexlimdvaacbv 44194. The equivalent of this theorem without the bound variable change is rexlimddv 3158. Usage of this theorem is discouraged because it depends on ax-13 2374, see rexlimddvcbvw 44195 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 44197* | 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 | rr-phpd 44198 | Equivalent of php 9244 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ ω) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | suceqd 44199 | Deduction associated with suceq 6451. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → suc 𝐴 = suc 𝐵) | ||
Theorem | tfindsd 44200* | Deduction associated with tfinds 7880. (Contributed by Rohan Ridenour, 8-Aug-2023.) |
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = suc 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂)) & ⊢ (𝜑 → 𝜒) & ⊢ ((𝜑 ∧ 𝑦 ∈ On ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ Lim 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝜃) → 𝜓) & ⊢ (𝜑 → 𝐴 ∈ On) ⇒ ⊢ (𝜑 → 𝜂) |
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