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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxkotf 21101* Functionality of function 𝑇. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       𝑇:(𝐾 × 𝑆)⟶𝒫 (𝑅 Cn 𝑆)
 
Theoremxkobval 21102* Alternative expression for the subbase of the compact-open topology. (Contributed by Mario Carneiro, 23-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ran 𝑇 = {𝑠 ∣ ∃𝑘 ∈ 𝒫 𝑋𝑣𝑆 ((𝑅t 𝑘) ∈ Comp ∧ 𝑠 = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})}
 
Theoremxkoval 21103* Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝑋 = 𝑅    &   𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅t 𝑥) ∈ Comp}    &   𝑇 = (𝑘𝐾, 𝑣𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓𝑘) ⊆ 𝑣})       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran 𝑇)))
 
Theoremxkotop 21104 The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top)
 
Theoremxkoopn 21105* 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} ∈ (𝑆 ^ko 𝑅))
 
Theoremtxtopi 21106 The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.)
𝑅 ∈ Top    &   𝑆 ∈ Top       (𝑅 ×t 𝑆) ∈ Top
 
Theoremtxtopon 21107 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 21108 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 21109 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 21110* The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
𝐽 = (∏t𝐹)       ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
 
Theoremptunimpt 21111* 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 21112* 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 21113 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 21114 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 21115 The base set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
𝐽 = (𝑆 ^ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = 𝐽)
 
Theoremxkotopon 21116 The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.)
𝐽 = (𝑆 ^ko 𝑅)       ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)))
 
Theoremptval2 21117* 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 21118 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 21119 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 21120 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 21121 Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
(((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
 
Theoremtxbasval 21122 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 21123 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 21124* 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 21125 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 21126 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 21127* 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 21128* 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 21129* 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 21130* 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 21131* 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 21132* 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 21133* Lemma for dfac14 21134. 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 21132 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 21134* Theorem ptcls 21132 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 21135* 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 (𝑆 ^ko 𝑅)))
 
Theoremtxcnp 21136* 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 21137* Lemma for ptcnp 21138. (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 21138* 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 21139* 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 21140* 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 21141* 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 21142 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 21143* 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 21144 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 21145 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶TopSp)       (𝜑𝑌 ∈ TopSp)
 
Theorempwstps 21146 A structure product of topologies is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ TopSp ∧ 𝐼𝑉) → 𝑌 ∈ TopSp)
 
Theoremtxrest 21147 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 21148 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 21149 Lemma for txindis 21150. (Contributed by Mario Carneiro, 14-Aug-2015.)
(( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
 
Theoremtxindis 21150 The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}
 
Theoremtxdis1cn 21151* 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 21152* 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 21153* 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 21154 The product of a collection of Hausdorff spaces is Hausdorff. (Contributed by Mario Carneiro, 2-Sep-2015.)
((𝐴𝑉𝐹:𝐴⟶Haus) → (∏t𝐹) ∈ Haus)
 
Theoremptrescn 21155* 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 21156* 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 21157* Lemma for txcmp 21159. (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 21158* Lemma for txcmp 21159. (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 21159 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 21160 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 21161 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.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
 
Theoremhauseqlcld 21162 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 21163 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 21164* 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 21165* 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 21166 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 21167 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 21168 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 21169 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) → (𝑆 ^ko 𝑅) ∈ Haus)
 
Theoremxkoptsub 21170 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)) ⊆ (𝑆 ^ko 𝑅))
 
Theoremxkopt 21171 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) → (𝑅 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝑅})))
 
Theoremxkopjcn 21172* 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 (𝑆 ^ko 𝑅) ×t 𝑅, but not without stronger assumptions on 𝑅; see xkofvcn 21200.) (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)) ∈ ((𝑆 ^ko 𝑅) Cn 𝑆))
 
Theoremxkoco1cn 21173* If 𝐹 is a continuous function, then 𝑔𝑔𝐹 is a continuous function on function spaces. (The reason we prove this and xkoco2cn 21174 independently of the more general xkococn 21176 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)) ∈ ((𝑇 ^ko 𝑆) Cn (𝑇 ^ko 𝑅)))
 
Theoremxkoco2cn 21174* 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)) ∈ ((𝑆 ^ko 𝑅) Cn (𝑇 ^ko 𝑅)))
 
Theoremxkococnlem 21175* 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 ∈ ((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅))(⟨𝐴, 𝐵⟩ ∈ 𝑧𝑧 ⊆ (𝐹 “ { ∈ (𝑅 Cn 𝑇) ∣ (𝐾) ⊆ 𝑉})))
 
Theoremxkococn 21176* 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 ∈ (((𝑇 ^ko 𝑆) ×t (𝑆 ^ko 𝑅)) Cn (𝑇 ^ko 𝑅)))
 
12.1.19  Continuous function-builders
 
Theoremcnmptid 21177* 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 21178* 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 21179* 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 21180* 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 21181* 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 21182* 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 21183* 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 21184* 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 21185* 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 21186* 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 21187* 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 21188* 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 21189* 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 21190* 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 21191* 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 21192* 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 21193* 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 21194* 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 21195* 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 (𝐽 ^ko 𝐾)))
 
Theoremcnmptkp 21196* 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 (𝐿 ^ko 𝐾)))    &   (𝜑𝐵𝑌)    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝜑 → (𝑥𝑋𝐶) ∈ (𝐽 Cn 𝐿))
 
Theoremcnmptk1 21197* 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 (𝐿 ^ko 𝐾)))    &   (𝜑 → (𝑧𝑍𝐵) ∈ (𝐿 Cn 𝑀))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
 
Theoremcnmpt1k 21198* 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 (𝑀 ^ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑦𝑌 ↦ (𝑥𝑋𝐶)) ∈ (𝐾 Cn (𝑀 ^ko 𝐽)))
 
Theoremcnmptkk 21199* 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 (𝐿 ^ko 𝐾)))    &   (𝜑 → (𝑥𝑋 ↦ (𝑧𝑍𝐵)) ∈ (𝐽 Cn (𝑀 ^ko 𝐿)))    &   (𝑧 = 𝐴𝐵 = 𝐶)       (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐶)) ∈ (𝐽 Cn (𝑀 ^ko 𝐾)))
 
Theoremxkofvcn 21200* Joint continuity of the function value operation as a function on continuous function spaces. (Compare xkopjcn 21172.) (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 ∈ (((𝑆 ^ko 𝑅) ×t 𝑅) Cn 𝑆))
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