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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxkoopn 22201* A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   (𝜑𝑅 ∈ Top)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑅t 𝐴) ∈ Comp)    &   (𝜑𝑈𝑆)       (𝜑 → {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝐴) ⊆ 𝑈} ∈ (𝑆ko 𝑅))

Theoremtxtopi 22202 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top       (𝑅 ×t 𝑆) ∈ Top

Theoremtxtopon 22203 The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Theoremtxuni 22204 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝑅    &   𝑌 = 𝑆       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))

Theoremtxunii 22205 The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top    &   𝑋 = 𝑅    &   𝑌 = 𝑆       (𝑋 × 𝑌) = (𝑅 ×t 𝑆)

Theoremptuni 22206* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)

Theoremptunimpt 22207* Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = 𝐽)

Theorempttopon 22208* The base set for the product topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝑥𝐴𝐾))       ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ (TopOn‘𝐵)) → 𝐽 ∈ (TopOn‘X𝑥𝐴 𝐵))

Theorempttoponconst 22209 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))       ((𝐴𝑉𝑅 ∈ (TopOn‘𝑋)) → 𝐽 ∈ (TopOn‘(𝑋m 𝐴)))

Theoremptuniconst 22210 The base set for a product topology when all factors are the same. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t‘(𝐴 × {𝑅}))    &   𝑋 = 𝑅       ((𝐴𝑉𝑅 ∈ Top) → (𝑋m 𝐴) = 𝐽)

Theoremxkouni 22211 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐽 = (𝑆ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)

Theoremxkotopon 22212 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (𝑆ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)))

Theoremptval2 22213* The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝐽 = (∏t𝐹)    &   𝑋 = 𝐽    &   𝐺 = (𝑘𝐴, 𝑢 ∈ (𝐹𝑘) ↦ ((𝑤𝑋 ↦ (𝑤𝑘)) “ 𝑢))       ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))

Theoremtxopn 22214 The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
(((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Theoremtxcld 22215 The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

Theoremtxcls 22216 Closure of a rectangle in the product topology. (Contributed by Mario Carneiro, 17-Sep-2015.)
(((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝐴𝑋𝐵𝑌)) → ((cls‘(𝑅 ×t 𝑆))‘(𝐴 × 𝐵)) = (((cls‘𝑅)‘𝐴) × ((cls‘𝑆)‘𝐵)))

Theoremtxss12 22217 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
(((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Theoremtxbasval 22218 It is sufficient to consider products of the bases for the topologies in the topological product. (Contributed by Mario Carneiro, 25-Aug-2014.)
((𝑅𝑉𝑆𝑊) → ((topGen‘𝑅) ×t (topGen‘𝑆)) = (𝑅 ×t 𝑆))

Theoremneitx 22219 The Cartesian product of two neighborhoods is a neighborhood in the product topology. (Contributed by Thierry Arnoux, 13-Jan-2018.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴 ∈ ((nei‘𝐽)‘𝐶) ∧ 𝐵 ∈ ((nei‘𝐾)‘𝐷))) → (𝐴 × 𝐵) ∈ ((nei‘(𝐽 ×t 𝐾))‘(𝐶 × 𝐷)))

Theoremtxcnpi 22220* Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))    &   (𝜑𝑈𝐿)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)       (𝜑 → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))

Theoremtx1cn 22221 Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))

Theoremtx2cn 22222 Continuity of the second projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Theoremptpjcn 22223* Continuity of a projection map into a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Feb-2015.)
𝑌 = 𝐽    &   𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) → (𝑥𝑌 ↦ (𝑥𝐼)) ∈ (𝐽 Cn (𝐹𝐼)))

Theoremptpjopn 22224* The projection map is an open map. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝑌 = 𝐽    &   𝐽 = (∏t𝐹)       (((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐼𝐴) ∧ 𝑈𝐽) → ((𝑥𝑌 ↦ (𝑥𝐼)) “ 𝑈) ∈ (𝐹𝐼))

Theoremptcld 22225* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶Top)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘(𝐹𝑘)))       (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t𝐹)))

Theoremptcldmpt 22226* A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐽 ∈ Top)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (Clsd‘𝐽))       (𝜑X𝑘𝐴 𝐶 ∈ (Clsd‘(∏t‘(𝑘𝐴𝐽))))

Theoremptclsg 22227* The closure of a box in the product topology is the box formed from the closures of the factors. The proof uses the axiom of choice; the last hypothesis is the choice assumption. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐽 = (∏t‘(𝑘𝐴𝑅))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝑆𝑋)    &   (𝜑 𝑘𝐴 𝑆AC 𝐴)       (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))

Theoremptcls 22228* The closure of a box in the product topology is the box formed from the closures of the factors. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (∏t‘(𝑘𝐴𝑅))    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑅 ∈ (TopOn‘𝑋))    &   ((𝜑𝑘𝐴) → 𝑆𝑋)       (𝜑 → ((cls‘𝐽)‘X𝑘𝐴 𝑆) = X𝑘𝐴 ((cls‘𝑅)‘𝑆))

Theoremdfac14lem 22229* Lemma for dfac14 22230. By equipping 𝑆 ∪ {𝑃} for some 𝑃𝑆 with the particular point topology, we can show that 𝑃 is in the closure of 𝑆; hence the sequence 𝑃(𝑥) is in the product of the closures, and we can utilize this instance of ptcls 22228 to extract an element of the closure of X𝑘𝐼𝑆. (Contributed by Mario Carneiro, 2-Sep-2015.)
(𝜑𝐼𝑉)    &   ((𝜑𝑥𝐼) → 𝑆𝑊)    &   ((𝜑𝑥𝐼) → 𝑆 ≠ ∅)    &   𝑃 = 𝒫 𝑆    &   𝑅 = {𝑦 ∈ 𝒫 (𝑆 ∪ {𝑃}) ∣ (𝑃𝑦𝑦 = (𝑆 ∪ {𝑃}))}    &   𝐽 = (∏t‘(𝑥𝐼𝑅))    &   (𝜑 → ((cls‘𝐽)‘X𝑥𝐼 𝑆) = X𝑥𝐼 ((cls‘𝑅)‘𝑆))       (𝜑X𝑥𝐼 𝑆 ≠ ∅)

Theoremdfac14 22230* Theorem ptcls 22228 is an equivalent of the axiom of choice. (Contributed by Mario Carneiro, 3-Sep-2015.)
(CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Top → ∀𝑠X 𝑘 ∈ dom 𝑓𝒫 (𝑓𝑘)((cls‘(∏t𝑓))‘X𝑘 ∈ dom 𝑓(𝑠𝑘)) = X𝑘 ∈ dom 𝑓((cls‘(𝑓𝑘))‘(𝑠𝑘))))

Theoremxkoccn 22231* The "constant function" function which maps 𝑥𝑌 to the constant function 𝑧𝑋𝑥 is a continuous function from 𝑋 into the space of continuous functions from 𝑌 to 𝑋. This can also be understood as the currying of the first projection function. (The currying of the second projection function is 𝑥𝑌 ↦ (𝑧𝑋𝑧), which we already know is continuous because it is a constant function.) (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑥𝑌 ↦ (𝑋 × {𝑥})) ∈ (𝑆 Cn (𝑆ko 𝑅)))

Theoremtxcnp 22232* If two functions are continuous at 𝐷, then the ordered pair of them is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐷𝑋)    &   (𝜑 → (𝑥𝑋𝐴) ∈ ((𝐽 CnP 𝐾)‘𝐷))    &   (𝜑 → (𝑥𝑋𝐵) ∈ ((𝐽 CnP 𝐿)‘𝐷))       (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 CnP (𝐾 ×t 𝐿))‘𝐷))

Theoremptcnplem 22233* Lemma for ptcnp 22234. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))    &   𝑘𝜓    &   ((𝜑𝜓) → 𝐺 Fn 𝐼)    &   (((𝜑𝜓) ∧ 𝑘𝐼) → (𝐺𝑘) ∈ (𝐹𝑘))    &   ((𝜑𝜓) → 𝑊 ∈ Fin)    &   (((𝜑𝜓) ∧ 𝑘 ∈ (𝐼𝑊)) → (𝐺𝑘) = (𝐹𝑘))    &   ((𝜑𝜓) → ((𝑥𝑋 ↦ (𝑘𝐼𝐴))‘𝐷) ∈ X𝑘𝐼 (𝐺𝑘))       ((𝜑𝜓) → ∃𝑧𝐽 (𝐷𝑧 ∧ ((𝑥𝑋 ↦ (𝑘𝐼𝐴)) “ 𝑧) ⊆ X𝑘𝐼 (𝐺𝑘)))

Theoremptcnp 22234* If every projection of a function is continuous at 𝐷, then the function itself is continuous at 𝐷 into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   (𝜑𝐷𝑋)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ ((𝐽 CnP (𝐹𝑘))‘𝐷))       (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷))

Theoremupxp 22235* Universal property of the Cartesian product considered as a categorical product in the category of sets. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝑃 = (1st ↾ (𝐵 × 𝐶))    &   𝑄 = (2nd ↾ (𝐵 × 𝐶))       ((𝐴𝐷𝐹:𝐴𝐵𝐺:𝐴𝐶) → ∃!(:𝐴⟶(𝐵 × 𝐶) ∧ 𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))

Theoremtxcnmpt 22236* A map into the product of two topological spaces is continuous if both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑊 = 𝑈    &   𝐻 = (𝑥𝑊 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → 𝐻 ∈ (𝑈 Cn (𝑅 ×t 𝑆)))

Theoremuptx 22237* Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑇 = (𝑅 ×t 𝑆)    &   𝑋 = 𝑅    &   𝑌 = 𝑆    &   𝑍 = (𝑋 × 𝑌)    &   𝑃 = (1st𝑍)    &   𝑄 = (2nd𝑍)       ((𝐹 ∈ (𝑈 Cn 𝑅) ∧ 𝐺 ∈ (𝑈 Cn 𝑆)) → ∃! ∈ (𝑈 Cn 𝑇)(𝐹 = (𝑃) ∧ 𝐺 = (𝑄)))

Theoremtxcn 22238 A map into the product of two topological spaces is continuous iff both of its projections are continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   𝑍 = (𝑋 × 𝑌)    &   𝑊 = 𝑈    &   𝑃 = (1st𝑍)    &   𝑄 = (2nd𝑍)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐹:𝑊𝑍) → (𝐹 ∈ (𝑈 Cn (𝑅 ×t 𝑆)) ↔ ((𝑃𝐹) ∈ (𝑈 Cn 𝑅) ∧ (𝑄𝐹) ∈ (𝑈 Cn 𝑆))))

Theoremptcn 22239* If every projection of a function is continuous, then the function itself is continuous into the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐾 = (∏t𝐹)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼⟶Top)    &   ((𝜑𝑘𝐼) → (𝑥𝑋𝐴) ∈ (𝐽 Cn (𝐹𝑘)))       (𝜑 → (𝑥𝑋 ↦ (𝑘𝐼𝐴)) ∈ (𝐽 Cn 𝐾))

Theoremprdstopn 22240 Topology of a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅 Fn 𝐼)    &   𝑂 = (TopOpen‘𝑌)       (𝜑𝑂 = (∏t‘(TopOpen ∘ 𝑅)))

Theoremprdstps 22241 A structure product of topological spaces is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶TopSp)       (𝜑𝑌 ∈ TopSp)

Theorempwstps 22242 A structure power of a topological space is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ TopSp ∧ 𝐼𝑉) → 𝑌 ∈ TopSp)

Theoremtxrest 22243 The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
(((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅t 𝐴) ×t (𝑆t 𝐵)))

Theoremtxdis 22244 The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Theoremtxindislem 22245 Lemma for txindis 22246. (Contributed by Mario Carneiro, 14-Aug-2015.)
(( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))

Theoremtxindis 22246 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Theoremtxdis1cn 22247* A function is jointly continuous on a discrete left topology iff it is continuous as a function of its right argument, for each fixed left value. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝜑𝑋𝑉)    &   (𝜑𝐽 ∈ (TopOn‘𝑌))    &   (𝜑𝐾 ∈ Top)    &   (𝜑𝐹 Fn (𝑋 × 𝑌))    &   ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑥𝐹𝑦)) ∈ (𝐽 Cn 𝐾))       (𝜑𝐹 ∈ ((𝒫 𝑋 ×t 𝐽) Cn 𝐾))

Theoremtxlly 22248* If the property 𝐴 is preserved under topological products, then so is the property of being locally 𝐴. (Contributed by Mario Carneiro, 10-Mar-2015.)
((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)       ((𝑅 ∈ Locally 𝐴𝑆 ∈ Locally 𝐴) → (𝑅 ×t 𝑆) ∈ Locally 𝐴)

Theoremtxnlly 22249* If the property 𝐴 is preserved under topological products, then so is the property of being n-locally 𝐴. (Contributed by Mario Carneiro, 13-Apr-2015.)
((𝑗𝐴𝑘𝐴) → (𝑗 ×t 𝑘) ∈ 𝐴)       ((𝑅 ∈ 𝑛-Locally 𝐴𝑆 ∈ 𝑛-Locally 𝐴) → (𝑅 ×t 𝑆) ∈ 𝑛-Locally 𝐴)

Theorempthaus 22250 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴𝑉𝐹:𝐴⟶Haus) → (∏t𝐹) ∈ Haus)

Theoremptrescn 22251* Restriction is a continuous function on product topologies. (Contributed by Mario Carneiro, 7-Feb-2015.)
𝑋 = 𝐽    &   𝐽 = (∏t𝐹)    &   𝐾 = (∏t‘(𝐹𝐵))       ((𝐴𝑉𝐹:𝐴⟶Top ∧ 𝐵𝐴) → (𝑥𝑋 ↦ (𝑥𝐵)) ∈ (𝐽 Cn 𝐾))

Theoremtxtube 22252* The "tube lemma". If 𝑋 is compact and there is an open set 𝑈 containing the line 𝑋 × {𝐴}, then there is a "tube" 𝑋 × 𝑢 for some neighborhood 𝑢 of 𝐴 which is entirely contained within 𝑈. (Contributed by Mario Carneiro, 21-Mar-2015.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Top)    &   (𝜑𝑈 ∈ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × {𝐴}) ⊆ 𝑈)    &   (𝜑𝐴𝑌)       (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ (𝑋 × 𝑢) ⊆ 𝑈))

Theoremtxcmplem1 22253* Lemma for txcmp 22255. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Comp)    &   (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × 𝑌) = 𝑊)    &   (𝜑𝐴𝑌)       (𝜑 → ∃𝑢𝑆 (𝐴𝑢 ∧ ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑢) ⊆ 𝑣))

Theoremtxcmplem2 22254* Lemma for txcmp 22255. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆    &   (𝜑𝑅 ∈ Comp)    &   (𝜑𝑆 ∈ Comp)    &   (𝜑𝑊 ⊆ (𝑅 ×t 𝑆))    &   (𝜑 → (𝑋 × 𝑌) = 𝑊)       (𝜑 → ∃𝑣 ∈ (𝒫 𝑊 ∩ Fin)(𝑋 × 𝑌) = 𝑣)

Theoremtxcmp 22255 The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)
((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Theoremtxcmpb 22256 The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝑋 = 𝑅    &   𝑌 = 𝑆       (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))

Theoremhausdiag 22257 A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))

Theoremhauseqlcld 22258 In a Hausdorff topology, the equalizer of two continuous functions is closed (thus, two continuous functions which agree on a dense set agree everywhere). (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐾 ∈ Haus)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → dom (𝐹𝐺) ∈ (Clsd‘𝐽))

Theoremtxhaus 22259 The topological product of two Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝑅 ∈ Haus ∧ 𝑆 ∈ Haus) → (𝑅 ×t 𝑆) ∈ Haus)

Theoremtxlm 22260* Two sequences converge iff the sequence of their ordered pairs converges. Proposition 14-2.6 of [Gleason] p. 230. (Contributed by NM, 16-Jul-2007.) (Revised by Mario Carneiro, 5-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑𝐺:𝑍𝑌)    &   𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)       (𝜑 → ((𝐹(⇝𝑡𝐽)𝑅𝐺(⇝𝑡𝐾)𝑆) ↔ 𝐻(⇝𝑡‘(𝐽 ×t 𝐾))⟨𝑅, 𝑆⟩))

Theoremlmcn2 22261* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑𝐺:𝑍𝑌)    &   (𝜑𝐹(⇝𝑡𝐽)𝑅)    &   (𝜑𝐺(⇝𝑡𝐾)𝑆)    &   (𝜑𝑂 ∈ ((𝐽 ×t 𝐾) Cn 𝑁))    &   𝐻 = (𝑛𝑍 ↦ ((𝐹𝑛)𝑂(𝐺𝑛)))       (𝜑𝐻(⇝𝑡𝑁)(𝑅𝑂𝑆))

Theoremtx1stc 22262 The topological product of two first-countable spaces is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 1stω ∧ 𝑆 ∈ 1stω) → (𝑅 ×t 𝑆) ∈ 1stω)

Theoremtx2ndc 22263 The topological product of two second-countable spaces is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
((𝑅 ∈ 2ndω ∧ 𝑆 ∈ 2ndω) → (𝑅 ×t 𝑆) ∈ 2ndω)

Theoremtxkgen 22264 The topological product of a locally compact space and a compactly generated Hausdorff space is compactly generated. (The condition on 𝑆 can also be replaced with either "compactly generated weak Hausdorff (CGWH)" or "compact Hausdorff-ly generated (CHG)", where WH means that all images of compact Hausdorff spaces are closed and CHG means that a set is open iff it is open in all compact Hausdorff spaces.) (Contributed by Mario Carneiro, 23-Mar-2015.)
((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ (ran 𝑘Gen ∩ Haus)) → (𝑅 ×t 𝑆) ∈ ran 𝑘Gen)

Theoremxkohaus 22265 If the codomain space is Hausdorff, then the compact-open topology of continuous functions is also Hausdorff. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Haus) → (𝑆ko 𝑅) ∈ Haus)

Theoremxkoptsub 22266 The compact-open topology is finer than the product topology restricted to continuous functions. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐽 = (∏t‘(𝑋 × {𝑆}))       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝐽t (𝑅 Cn 𝑆)) ⊆ (𝑆ko 𝑅))

Theoremxkopt 22267 The compact-open topology on a discrete set coincides with the product topology where all the factors are the same. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
((𝑅 ∈ Top ∧ 𝐴𝑉) → (𝑅ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))

Theoremxkopjcn 22268* Continuity of a projection map from the space of continuous functions. (This theorem can be strengthened, to joint continuity in both 𝑓 and 𝐴 as a function on (𝑆ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 22296.) (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top ∧ 𝐴𝑋) → (𝑓 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝐴)) ∈ ((𝑆ko 𝑅) Cn 𝑆))

Theoremxkoco1cn 22269* If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 22270 independently of the more general xkococn 22272 is because that requires some inconvenient extra assumptions on 𝑆.) (Contributed by Mario Carneiro, 20-Mar-2015.)
(𝜑𝑇 ∈ Top)    &   (𝜑𝐹 ∈ (𝑅 Cn 𝑆))       (𝜑 → (𝑔 ∈ (𝑆 Cn 𝑇) ↦ (𝑔𝐹)) ∈ ((𝑇ko 𝑆) Cn (𝑇ko 𝑅)))

Theoremxkoco2cn 22270* If 𝐹 is a continuous function, then 𝑔𝐹𝑔 is a continuous function on function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝑅 ∈ Top)    &   (𝜑𝐹 ∈ (𝑆 Cn 𝑇))       (𝜑 → (𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝐹𝑔)) ∈ ((𝑆ko 𝑅) Cn (𝑇ko 𝑅)))

Theoremxkococnlem 22271* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))    &   (𝜑𝑆 ∈ 𝑛-Locally Comp)    &   (𝜑𝐾 𝑅)    &   (𝜑 → (𝑅t 𝐾) ∈ Comp)    &   (𝜑𝑉𝑇)    &   (𝜑𝐴 ∈ (𝑆 Cn 𝑇))    &   (𝜑𝐵 ∈ (𝑅 Cn 𝑆))    &   (𝜑 → ((𝐴𝐵) “ 𝐾) ⊆ 𝑉)       (𝜑 → ∃𝑧 ∈ ((𝑇ko 𝑆) ×t (𝑆ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))

Theoremxkococn 22272* Continuity of the composition operation as a function on continuous function spaces. (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝐹 = (𝑓 ∈ (𝑆 Cn 𝑇), 𝑔 ∈ (𝑅 Cn 𝑆) ↦ (𝑓𝑔))       ((𝑅 ∈ Top ∧ 𝑆 ∈ 𝑛-Locally Comp ∧ 𝑇 ∈ Top) → 𝐹 ∈ (((𝑇ko 𝑆) ×t (𝑆ko 𝑅)) Cn (𝑇ko 𝑅)))

12.1.19  Continuous function-builders

Theoremcnmptid 22273* The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))       (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))

Theoremcnmptc 22274* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝑃𝑌)       (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))

Theoremcnmpt11 22275* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑦𝑌𝐵) ∈ (𝐾 Cn 𝐿))    &   (𝑦 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))

Theoremcnmpt11f 22276* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐹 ∈ (𝐾 Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))

Theoremcnmpt1t 22277* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐿)))

Theoremcnmpt12f 22278* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))    &   (𝜑𝐹 ∈ ((𝐾 ×t 𝐿) Cn 𝑀))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝐽 Cn 𝑀))

Theoremcnmpt12 22279* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 12-Jun-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐿))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑦𝑌, 𝑧𝑍𝐶) ∈ ((𝐾 ×t 𝐿) Cn 𝑀))    &   ((𝑦 = 𝐴𝑧 = 𝐵) → 𝐶 = 𝐷)       (𝜑 → (𝑥𝑋𝐷) ∈ (𝐽 Cn 𝑀))

Theoremcnmpt1st 22280* The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))

Theoremcnmpt2nd 22281* The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))

Theoremcnmpt2c 22282* A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑃𝑍)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Theoremcnmpt21 22283* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐶) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Theoremcnmpt21f 22284* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑𝐹 ∈ (𝐿 Cn 𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐹𝐴)) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Theoremcnmpt2t 22285* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))

Theoremcnmpt22 22286* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑 → (𝑧𝑍, 𝑤𝑊𝐶) ∈ ((𝐿 ×t 𝑀) Cn 𝑁))    &   ((𝑧 = 𝐴𝑤 = 𝐵) → 𝐶 = 𝐷)       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐷) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Theoremcnmpt22f 22287* The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))    &   (𝜑𝐹 ∈ ((𝐿 ×t 𝑀) Cn 𝑁))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐹𝐵)) ∈ ((𝐽 ×t 𝐾) Cn 𝑁))

Theoremcnmpt1res 22288* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 5-Jun-2014.)
𝐾 = (𝐽t 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))       (𝜑 → (𝑥𝑌𝐴) ∈ (𝐾 Cn 𝐿))

Theoremcnmpt2res 22289* The restriction of a continuous function to a subset is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝐾 = (𝐽t 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑁 = (𝑀t 𝑊)    &   (𝜑𝑀 ∈ (TopOn‘𝑍))    &   (𝜑𝑊𝑍)    &   (𝜑 → (𝑥𝑋, 𝑦𝑍𝐴) ∈ ((𝐽 ×t 𝑀) Cn 𝐿))       (𝜑 → (𝑥𝑌, 𝑦𝑊𝐴) ∈ ((𝐾 ×t 𝑁) Cn 𝐿))

Theoremcnmptcom 22290* The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))       (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))

Theoremcnmptkc 22291* The curried first projection function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝑥)) ∈ (𝐽 Cn (𝐽ko 𝐾)))

Theoremcnmptkp 22292* The evaluation of the inner function in a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑𝐵𝑌)    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))

Theoremcnmptk1 22293* The composition of a curried function with a one-arg function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))

Theoremcnmpt1k 22294* The composition of a one-arg function with a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐿))    &   (𝜑 → (𝑦𝑌 ↦ (𝑧𝑍𝐵)) ∈ (𝐾 Cn (𝑀ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀ko 𝐽)))

Theoremcnmptkk 22295* The composition of two curried functions is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝑀 ∈ (TopOn‘𝑊))    &   (𝜑𝐿 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀ko 𝐾)))

Theoremxkofvcn 22296* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 22268.) (Contributed by Mario Carneiro, 20-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
𝑋 = 𝑅    &   𝐹 = (𝑓 ∈ (𝑅 Cn 𝑆), 𝑥𝑋 ↦ (𝑓𝑥))       ((𝑅 ∈ 𝑛-Locally Comp ∧ 𝑆 ∈ Top) → 𝐹 ∈ (((𝑆ko 𝑅) ×t 𝑅) Cn 𝑆))

Theoremcnmptk1p 22297* The evaluation of a curried function by a one-arg function is jointly continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))

Theoremcnmptk2 22298* The uncurrying of a curried function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (TopOn‘𝑍))    &   (𝜑𝐾 ∈ 𝑛-Locally Comp)    &   (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))       (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Theoremxkoinjcn 22299* Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝐹 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩))       ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×t 𝑅) ↑ko 𝑆)))

Theoremcnmpt2k 22300* The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))

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