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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddcnlem 24601* Lemma for addcn 24602, subcn 24603, and mulcn 24604. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpenβ€˜β„‚fld)    &    + :(β„‚ Γ— β„‚)βŸΆβ„‚    &   ((π‘Ž ∈ ℝ+ ∧ 𝑏 ∈ β„‚ ∧ 𝑐 ∈ β„‚) β†’ βˆƒπ‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘’ ∈ β„‚ βˆ€π‘£ ∈ β„‚ (((absβ€˜(𝑒 βˆ’ 𝑏)) < 𝑦 ∧ (absβ€˜(𝑣 βˆ’ 𝑐)) < 𝑧) β†’ (absβ€˜((𝑒 + 𝑣) βˆ’ (𝑏 + 𝑐))) < π‘Ž))    β‡’    + ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremaddcn 24602 Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’    + ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremsubcn 24603 Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’    βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremmulcn 24604 Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by NM, 30-Jul-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’    Β· ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
TheoremdivcnOLD 24605 Obsolete version of divcn 24607 as of 6-Apr-2025. (Contributed by Mario Carneiro, 12-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΎ = (𝐽 β†Ύt (β„‚ βˆ– {0}))    β‡’    / ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽)
 
Theoremmpomulcn 24606* Complex number multiplication is a continuous function. Version of mulcn 24604 using maps-to notation, which does not require ax-mulf 11193. (Contributed by GG, 16-Mar-2025.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   (π‘₯ ∈ β„‚, 𝑦 ∈ β„‚ ↦ (π‘₯ Β· 𝑦)) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremdivcn 24607 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΎ = (𝐽 β†Ύt (β„‚ βˆ– {0}))    β‡’    / ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽)
 
Theoremcnfldtgp 24608 The complex numbers form a topological group under addition, with the standard topology induced by the absolute value metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
β„‚fld ∈ TopGrp
 
Theoremfsumcn 24609* A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐡 normally contains free variables π‘˜ and π‘₯ to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
𝐾 = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝐽 Cn 𝐾))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ Ξ£π‘˜ ∈ 𝐴 𝐡) ∈ (𝐽 Cn 𝐾))
 
Theoremfsum2cn 24610* Version of fsumcn 24609 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
𝐾 = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐿 ∈ (TopOnβ€˜π‘Œ))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝐡) ∈ ((𝐽 Γ—t 𝐿) Cn 𝐾))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ Ξ£π‘˜ ∈ 𝐴 𝐡) ∈ ((𝐽 Γ—t 𝐿) Cn 𝐾))
 
Theoremexpcn 24611* The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   (𝑁 ∈ β„•0 β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
Theoremdivccn 24612* Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯ / 𝐴)) ∈ (𝐽 Cn 𝐽))
 
TheoremexpcnOLD 24613* Obsolete version of expcn 24611 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   (𝑁 ∈ β„•0 β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯↑𝑁)) ∈ (𝐽 Cn 𝐽))
 
TheoremdivccnOLD 24614* Obsolete version of divccn 24612 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯ / 𝐴)) ∈ (𝐽 Cn 𝐽))
 
Theoremsqcn 24615* The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   (π‘₯ ∈ β„‚ ↦ (π‘₯↑2)) ∈ (𝐽 Cn 𝐽)
 
12.4.11  Topological definitions using the reals
 
Syntaxcii 24616 Extend class notation with the unit interval.
class II
 
Syntaxccncf 24617 Extend class notation to include the operation which returns a class of continuous complex functions.
class –cnβ†’
 
Definitiondf-ii 24618 Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
II = (MetOpenβ€˜((abs ∘ βˆ’ ) β†Ύ ((0[,]1) Γ— (0[,]1))))
 
Definitiondf-cncf 24619* Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
–cnβ†’ = (π‘Ž ∈ 𝒫 β„‚, 𝑏 ∈ 𝒫 β„‚ ↦ {𝑓 ∈ (𝑏 ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘’ ∈ ℝ+ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘¦ ∈ π‘Ž ((absβ€˜(π‘₯ βˆ’ 𝑦)) < 𝑑 β†’ (absβ€˜((π‘“β€˜π‘₯) βˆ’ (π‘“β€˜π‘¦))) < 𝑒)})
 
Theoremiitopon 24620 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
II ∈ (TopOnβ€˜(0[,]1))
 
Theoremiitop 24621 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
II ∈ Top
 
Theoremiiuni 24622 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
(0[,]1) = βˆͺ II
 
Theoremdfii2 24623 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
II = ((topGenβ€˜ran (,)) β†Ύt (0[,]1))
 
Theoremdfii3 24624 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   II = (𝐽 β†Ύt (0[,]1))
 
Theoremdfii4 24625 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐼 = (β„‚fld β†Ύs (0[,]1))    β‡’   II = (TopOpenβ€˜πΌ)
 
Theoremdfii5 24626 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
II = (ordTopβ€˜( ≀ ∩ ((0[,]1) Γ— (0[,]1))))
 
Theoremiicmp 24627 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp
 
Theoremiiconn 24628 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn
 
Theoremcncfval 24629* The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐡. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ (𝐴–cn→𝐡) = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝐴 ((absβ€˜(π‘₯ βˆ’ 𝑀)) < 𝑧 β†’ (absβ€˜((π‘“β€˜π‘₯) βˆ’ (π‘“β€˜π‘€))) < 𝑦)})
 
Theoremelcncf 24630* Membership in the set of continuous complex functions from 𝐴 to 𝐡. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ (𝐹 ∈ (𝐴–cn→𝐡) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝐴 ((absβ€˜(π‘₯ βˆ’ 𝑀)) < 𝑧 β†’ (absβ€˜((πΉβ€˜π‘₯) βˆ’ (πΉβ€˜π‘€))) < 𝑦))))
 
Theoremelcncf2 24631* Version of elcncf 24630 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ (𝐹 ∈ (𝐴–cn→𝐡) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝐴 ((absβ€˜(𝑀 βˆ’ π‘₯)) < 𝑧 β†’ (absβ€˜((πΉβ€˜π‘€) βˆ’ (πΉβ€˜π‘₯))) < 𝑦))))
 
Theoremcncfrss 24632 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴–cn→𝐡) β†’ 𝐴 βŠ† β„‚)
 
Theoremcncfrss2 24633 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴–cn→𝐡) β†’ 𝐡 βŠ† β„‚)
 
Theoremcncff 24634 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴–cn→𝐡) β†’ 𝐹:𝐴⟢𝐡)
 
Theoremcncfi 24635* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴–cn→𝐡) ∧ 𝐢 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) β†’ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝐴 ((absβ€˜(𝑀 βˆ’ 𝐢)) < 𝑧 β†’ (absβ€˜((πΉβ€˜π‘€) βˆ’ (πΉβ€˜πΆ))) < 𝑅))
 
Theoremelcncf1di 24636* Membership in the set of continuous complex functions from 𝐴 to 𝐡. (Contributed by Paul Chapman, 26-Nov-2007.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) β†’ 𝑍 ∈ ℝ+))    &   (πœ‘ β†’ (((π‘₯ ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) β†’ ((absβ€˜(π‘₯ βˆ’ 𝑀)) < 𝑍 β†’ (absβ€˜((πΉβ€˜π‘₯) βˆ’ (πΉβ€˜π‘€))) < 𝑦)))    β‡’   (πœ‘ β†’ ((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ 𝐹 ∈ (𝐴–cn→𝐡)))
 
Theoremelcncf1ii 24637* Membership in the set of continuous complex functions from 𝐴 to 𝐡. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴⟢𝐡    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) β†’ 𝑍 ∈ ℝ+)    &   (((π‘₯ ∈ 𝐴 ∧ 𝑀 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) β†’ ((absβ€˜(π‘₯ βˆ’ 𝑀)) < 𝑍 β†’ (absβ€˜((πΉβ€˜π‘₯) βˆ’ (πΉβ€˜π‘€))) < 𝑦))    β‡’   ((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ 𝐹 ∈ (𝐴–cn→𝐡))
 
Theoremrescncf 24638 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐢 βŠ† 𝐴 β†’ (𝐹 ∈ (𝐴–cn→𝐡) β†’ (𝐹 β†Ύ 𝐢) ∈ (𝐢–cn→𝐡)))
 
Theoremcncfcdm 24639 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
((𝐢 βŠ† β„‚ ∧ 𝐹 ∈ (𝐴–cn→𝐡)) β†’ (𝐹 ∈ (𝐴–cn→𝐢) ↔ 𝐹:𝐴⟢𝐢))
 
Theoremcncfss 24640 The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐡 βŠ† 𝐢 ∧ 𝐢 βŠ† β„‚) β†’ (𝐴–cn→𝐡) βŠ† (𝐴–cn→𝐢))
 
Theoremclimcncf 24641 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cn→𝐡))    &   (πœ‘ β†’ 𝐺:π‘βŸΆπ΄)    &   (πœ‘ β†’ 𝐺 ⇝ 𝐷)    &   (πœ‘ β†’ 𝐷 ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝐹 ∘ 𝐺) ⇝ (πΉβ€˜π·))
 
Theoremabscncf 24642 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremrecncf 24643 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
β„œ ∈ (ℂ–cn→ℝ)
 
Theoremimcncf 24644 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
β„‘ ∈ (ℂ–cn→ℝ)
 
Theoremcjcncf 24645 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
βˆ— ∈ (ℂ–cnβ†’β„‚)
 
Theoremmulc1cncf 24646* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (𝐴 Β· π‘₯))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremdivccncf 24647* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ / 𝐴))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremcncfco 24648 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
(πœ‘ β†’ 𝐹 ∈ (𝐴–cn→𝐡))    &   (πœ‘ β†’ 𝐺 ∈ (𝐡–cn→𝐢))    β‡’   (πœ‘ β†’ (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐢))
 
Theoremcncfcompt2 24649* Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘₯πœ‘    &   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐡))    &   (πœ‘ β†’ (𝑦 ∈ 𝐢 ↦ 𝑆) ∈ (𝐢–cn→𝐸))    &   (πœ‘ β†’ 𝐡 βŠ† 𝐢)    &   (𝑦 = 𝑅 β†’ 𝑆 = 𝑇)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸))
 
Theoremcncfmet 24650 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
𝐢 = ((abs ∘ βˆ’ ) β†Ύ (𝐴 Γ— 𝐴))    &   π· = ((abs ∘ βˆ’ ) β†Ύ (𝐡 Γ— 𝐡))    &   π½ = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ (𝐴–cn→𝐡) = (𝐽 Cn 𝐾))
 
Theoremcncfcn 24651 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΎ = (𝐽 β†Ύt 𝐴)    &   πΏ = (𝐽 β†Ύt 𝐡)    β‡’   ((𝐴 βŠ† β„‚ ∧ 𝐡 βŠ† β„‚) β†’ (𝐴–cn→𝐡) = (𝐾 Cn 𝐿))
 
Theoremcncfcn1 24652 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
𝐽 = (TopOpenβ€˜β„‚fld)    β‡’   (ℂ–cnβ†’β„‚) = (𝐽 Cn 𝐽)
 
Theoremcncfmptc 24653* A constant function is a continuous function on β„‚. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
((𝐴 ∈ 𝑇 ∧ 𝑆 βŠ† β„‚ ∧ 𝑇 βŠ† β„‚) β†’ (π‘₯ ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇))
 
Theoremcncfmptid 24654* The identity function is a continuous function on β„‚. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
((𝑆 βŠ† 𝑇 ∧ 𝑇 βŠ† β„‚) β†’ (π‘₯ ∈ 𝑆 ↦ π‘₯) ∈ (𝑆–cn→𝑇))
 
Theoremcncfmpt1f 24655* Composition of continuous functions. –cnβ†’ analogue of cnmpt11f 23389. (Contributed by Mario Carneiro, 3-Sep-2014.)
(πœ‘ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (πΉβ€˜π΄)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremcncfmpt2f 24656* Composition of continuous functions. –cnβ†’ analogue of cnmpt12f 23391. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremcncfmpt2ss 24657* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΉ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cn→𝑆))    &   π‘† βŠ† β„‚    &   ((𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴𝐹𝐡) ∈ 𝑆)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴𝐹𝐡)) ∈ (𝑋–cn→𝑆))
 
Theoremaddccncf 24658* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ + 𝐴))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremidcncf 24659 The identity function is a continuous function on β„‚. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 24654 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐹 = (π‘₯ ∈ β„‚ ↦ π‘₯)    β‡’   πΉ ∈ (ℂ–cnβ†’β„‚)
 
Theoremsub1cncf 24660* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ βˆ’ 𝐴))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremsub2cncf 24661* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (𝐴 βˆ’ π‘₯))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremcdivcncf 24662* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
𝐹 = (π‘₯ ∈ (β„‚ βˆ– {0}) ↦ (𝐴 / π‘₯))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ ((β„‚ βˆ– {0})–cnβ†’β„‚))
 
Theoremnegcncf 24663* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ -π‘₯)    β‡’   (𝐴 βŠ† β„‚ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))
 
TheoremnegcncfOLD 24664* Obsolete version of negcncf 24663 as of 9-Apr-2025. (Contributed by Mario Carneiro, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ -π‘₯)    β‡’   (𝐴 βŠ† β„‚ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))
 
Theoremnegfcncf 24665* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
𝐺 = (π‘₯ ∈ 𝐴 ↦ -(πΉβ€˜π‘₯))    β‡’   (𝐹 ∈ (𝐴–cnβ†’β„‚) β†’ 𝐺 ∈ (𝐴–cnβ†’β„‚))
 
TheoremabscncfALT 24666 Absolute value is continuous. Alternate proof of abscncf 24642. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremcncfcnvcn 24667 Rewrite cmphaushmeo 23525 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΎ = (𝐽 β†Ύt 𝑋)    β‡’   ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cnβ†’π‘Œ)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ ◑𝐹 ∈ (π‘Œβ€“cn→𝑋)))
 
Theoremexpcncf 24668* The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 24611. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ β„•0 β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯↑𝑁)) ∈ (ℂ–cnβ†’β„‚))
 
Theoremcnmptre 24669* Lemma for iirevcn 24672 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
𝑅 = (TopOpenβ€˜β„‚fld)    &   π½ = ((topGenβ€˜ran (,)) β†Ύt 𝐴)    &   πΎ = ((topGenβ€˜ran (,)) β†Ύt 𝐡)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ 𝐡 βŠ† ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ (π‘₯ ∈ β„‚ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))
 
Theoremcnmpopc 24670* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
𝑅 = (topGenβ€˜ran (,))    &   π‘€ = (𝑅 β†Ύt (𝐴[,]𝐡))    &   π‘ = (𝑅 β†Ύt (𝐡[,]𝐢))    &   π‘‚ = (𝑅 β†Ύt (𝐴[,]𝐢))    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴[,]𝐢))    &   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))    &   ((πœ‘ ∧ (π‘₯ = 𝐡 ∧ 𝑦 ∈ 𝑋)) β†’ 𝐷 = 𝐸)    &   (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐡), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 Γ—t 𝐽) Cn 𝐾))    &   (πœ‘ β†’ (π‘₯ ∈ (𝐡[,]𝐢), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 Γ—t 𝐽) Cn 𝐾))    β‡’   (πœ‘ β†’ (π‘₯ ∈ (𝐴[,]𝐢), 𝑦 ∈ 𝑋 ↦ if(π‘₯ ≀ 𝐡, 𝐷, 𝐸)) ∈ ((𝑂 Γ—t 𝐽) Cn 𝐾))
 
Theoremiirev 24671 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,]1) β†’ (1 βˆ’ 𝑋) ∈ (0[,]1))
 
Theoremiirevcn 24672 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
(π‘₯ ∈ (0[,]1) ↦ (1 βˆ’ π‘₯)) ∈ (II Cn II)
 
Theoremiihalf1 24673 Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ (0[,](1 / 2)) β†’ (2 Β· 𝑋) ∈ (0[,]1))
 
Theoremiihalf1cn 24674 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGenβ€˜ran (,)) β†Ύt (0[,](1 / 2)))    β‡’   (π‘₯ ∈ (0[,](1 / 2)) ↦ (2 Β· π‘₯)) ∈ (𝐽 Cn II)
 
Theoremiihalf1cnOLD 24675 Obsolete version of iihalf1cn 24674 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = ((topGenβ€˜ran (,)) β†Ύt (0[,](1 / 2)))    β‡’   (π‘₯ ∈ (0[,](1 / 2)) ↦ (2 Β· π‘₯)) ∈ (𝐽 Cn II)
 
Theoremiihalf2 24676 Map the second half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑋 ∈ ((1 / 2)[,]1) β†’ ((2 Β· 𝑋) βˆ’ 1) ∈ (0[,]1))
 
Theoremiihalf2cn 24677 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = ((topGenβ€˜ran (,)) β†Ύt ((1 / 2)[,]1))    β‡’   (π‘₯ ∈ ((1 / 2)[,]1) ↦ ((2 Β· π‘₯) βˆ’ 1)) ∈ (𝐽 Cn II)
 
Theoremiihalf2cnOLD 24678 Obsolete version of iihalf2cn 24677 as of 9-Apr-2025. (Contributed by Mario Carneiro, 6-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = ((topGenβ€˜ran (,)) β†Ύt ((1 / 2)[,]1))    β‡’   (π‘₯ ∈ ((1 / 2)[,]1) ↦ ((2 Β· π‘₯) βˆ’ 1)) ∈ (𝐽 Cn II)
 
Theoremelii1 24679 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
(𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≀ (1 / 2)))
 
Theoremelii2 24680 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
((𝑋 ∈ (0[,]1) ∧ Β¬ 𝑋 ≀ (1 / 2)) β†’ 𝑋 ∈ ((1 / 2)[,]1))
 
Theoremiimulcl 24681 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ (0[,]1) ∧ 𝐡 ∈ (0[,]1)) β†’ (𝐴 Β· 𝐡) ∈ (0[,]1))
 
Theoremiimulcn 24682* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
(π‘₯ ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (π‘₯ Β· 𝑦)) ∈ ((II Γ—t II) Cn II)
 
TheoremiimulcnOLD 24683* Obsolete version of iimulcn 24682 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(π‘₯ ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (π‘₯ Β· 𝑦)) ∈ ((II Γ—t II) Cn II)
 
Theoremicoopnst 24684 A half-open interval starting at 𝐴 is open in the closed interval from 𝐴 to 𝐡. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝐽 = (MetOpenβ€˜((abs ∘ βˆ’ ) β†Ύ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐢 ∈ (𝐴(,]𝐡) β†’ (𝐴[,)𝐢) ∈ 𝐽))
 
Theoremiocopnst 24685 A half-open interval ending at 𝐡 is open in the closed interval from 𝐴 to 𝐡. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝐽 = (MetOpenβ€˜((abs ∘ βˆ’ ) β†Ύ ((𝐴[,]𝐡) Γ— (𝐴[,]𝐡))))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐢 ∈ (𝐴[,)𝐡) β†’ (𝐢(,]𝐡) ∈ 𝐽))
 
Theoremicchmeo 24686* The natural bijection from [0, 1] to an arbitrary nontrivial closed interval [𝐴, 𝐡] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΉ = (π‘₯ ∈ (0[,]1) ↦ ((π‘₯ Β· 𝐡) + ((1 βˆ’ π‘₯) Β· 𝐴)))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ 𝐹 ∈ (IIHomeo(𝐽 β†Ύt (𝐴[,]𝐡))))
 
TheoremicchmeoOLD 24687* Obsolete version of icchmeo 24686 as of 9-Apr-2025. (Contributed by Mario Carneiro, 8-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐽 = (TopOpenβ€˜β„‚fld)    &   πΉ = (π‘₯ ∈ (0[,]1) ↦ ((π‘₯ Β· 𝐡) + ((1 βˆ’ π‘₯) Β· 𝐴)))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡) β†’ 𝐹 ∈ (IIHomeo(𝐽 β†Ύt (𝐴[,]𝐡))))
 
Theoremicopnfcnv 24688* Define a bijection from [0, 1) to [0, +∞). (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,)1) ↦ (π‘₯ / (1 βˆ’ π‘₯)))    β‡’   (𝐹:(0[,)1)–1-1-ontoβ†’(0[,)+∞) ∧ ◑𝐹 = (𝑦 ∈ (0[,)+∞) ↦ (𝑦 / (1 + 𝑦))))
 
Theoremicopnfhmeo 24689* The defined bijection from [0, 1) to [0, +∞) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,)1) ↦ (π‘₯ / (1 βˆ’ π‘₯)))    &   π½ = (TopOpenβ€˜β„‚fld)    β‡’   (𝐹 Isom < , < ((0[,)1), (0[,)+∞)) ∧ 𝐹 ∈ ((𝐽 β†Ύt (0[,)1))Homeo(𝐽 β†Ύt (0[,)+∞))))
 
Theoremiccpnfcnv 24690* Define a bijection from [0, 1] to [0, +∞]. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,]1) ↦ if(π‘₯ = 1, +∞, (π‘₯ / (1 βˆ’ π‘₯))))    β‡’   (𝐹:(0[,]1)–1-1-ontoβ†’(0[,]+∞) ∧ ◑𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
 
Theoremiccpnfhmeo 24691 The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,]1) ↦ if(π‘₯ = 1, +∞, (π‘₯ / (1 βˆ’ π‘₯))))    &   πΎ = ((ordTopβ€˜ ≀ ) β†Ύt (0[,]+∞))    β‡’   (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))
 
Theoremxrhmeo 24692* The bijection from [-1, 1] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐹 = (π‘₯ ∈ (0[,]1) ↦ if(π‘₯ = 1, +∞, (π‘₯ / (1 βˆ’ π‘₯))))    &   πΊ = (𝑦 ∈ (-1[,]1) ↦ if(0 ≀ 𝑦, (πΉβ€˜π‘¦), -𝑒(πΉβ€˜-𝑦)))    &   π½ = (TopOpenβ€˜β„‚fld)    β‡’   (𝐺 Isom < , < ((-1[,]1), ℝ*) ∧ 𝐺 ∈ ((𝐽 β†Ύt (-1[,]1))Homeo(ordTopβ€˜ ≀ )))
 
Theoremxrhmph 24693 The extended reals are homeomorphic to the interval [0, 1]. (Contributed by Mario Carneiro, 9-Sep-2015.)
II ≃ (ordTopβ€˜ ≀ )
 
Theoremxrcmp 24694 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 24543), this means that ℝ* is a compactification of ℝ. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTopβ€˜ ≀ ) ∈ Comp
 
Theoremxrconn 24695 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
(ordTopβ€˜ ≀ ) ∈ Conn
 
Theoremicccvx 24696 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((𝐢 ∈ (𝐴[,]𝐡) ∧ 𝐷 ∈ (𝐴[,]𝐡) ∧ 𝑇 ∈ (0[,]1)) β†’ (((1 βˆ’ 𝑇) Β· 𝐢) + (𝑇 Β· 𝐷)) ∈ (𝐴[,]𝐡)))
 
Theoremoprpiece1res1 24697* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    &   π΄ ≀ 𝐡    &   π‘… ∈ V    &   π‘† ∈ V    &   πΎ ∈ (𝐴[,]𝐡)    &   πΉ = (π‘₯ ∈ (𝐴[,]𝐡), 𝑦 ∈ 𝐢 ↦ if(π‘₯ ≀ 𝐾, 𝑅, 𝑆))    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐾), 𝑦 ∈ 𝐢 ↦ 𝑅)    β‡’   (𝐹 β†Ύ ((𝐴[,]𝐾) Γ— 𝐢)) = 𝐺
 
Theoremoprpiece1res2 24698* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
𝐴 ∈ ℝ    &   π΅ ∈ ℝ    &   π΄ ≀ 𝐡    &   π‘… ∈ V    &   π‘† ∈ V    &   πΎ ∈ (𝐴[,]𝐡)    &   πΉ = (π‘₯ ∈ (𝐴[,]𝐡), 𝑦 ∈ 𝐢 ↦ if(π‘₯ ≀ 𝐾, 𝑅, 𝑆))    &   (π‘₯ = 𝐾 β†’ 𝑅 = 𝑃)    &   (π‘₯ = 𝐾 β†’ 𝑆 = 𝑄)    &   (𝑦 ∈ 𝐢 β†’ 𝑃 = 𝑄)    &   πΊ = (π‘₯ ∈ (𝐾[,]𝐡), 𝑦 ∈ 𝐢 ↦ 𝑆)    β‡’   (𝐹 β†Ύ ((𝐾[,]𝐡) Γ— 𝐢)) = 𝐺
 
Theoremcnrehmeo 24699* The canonical bijection from (ℝ Γ— ℝ) to β„‚ described in cnref1o 12974 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if (ℝ Γ— ℝ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.) Avoid ax-mulf 11193. (Revised by GG, 16-Mar-2025.)
𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (π‘₯ + (i Β· 𝑦)))    &   π½ = (topGenβ€˜ran (,))    &   πΎ = (TopOpenβ€˜β„‚fld)    β‡’   πΉ ∈ ((𝐽 Γ—t 𝐽)Homeo𝐾)
 
TheoremcnrehmeoOLD 24700* Obsolete version of cnrehmeo 24699 as of 9-Apr-2025. (Contributed by Mario Carneiro, 25-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (π‘₯ + (i Β· 𝑦)))    &   π½ = (topGenβ€˜ran (,))    &   πΎ = (TopOpenβ€˜β„‚fld)    β‡’   πΉ ∈ ((𝐽 Γ—t 𝐽)Homeo𝐾)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47941
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