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Theorem List for Intuitionistic Logic Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetss2 14401* If the metric 𝐷 is "strongly finer" than 𝐢 (meaning that there is a positive real constant 𝑅 such that 𝐢(π‘₯, 𝑦) ≀ 𝑅 Β· 𝐷(π‘₯, 𝑦)), then 𝐷 generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    &   (πœ‘ β†’ 𝐢 ∈ (Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝐷 ∈ (Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑅 ∈ ℝ+)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (π‘₯𝐢𝑦) ≀ (𝑅 Β· (π‘₯𝐷𝑦)))    β‡’   (πœ‘ β†’ 𝐽 βŠ† 𝐾)
 
Theoremcomet 14402* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
(πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝐹:(0[,]+∞)βŸΆβ„*)    &   ((πœ‘ ∧ π‘₯ ∈ (0[,]+∞)) β†’ ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0))    &   ((πœ‘ ∧ (π‘₯ ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) β†’ (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≀ (πΉβ€˜π‘¦)))    &   ((πœ‘ ∧ (π‘₯ ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) β†’ (πΉβ€˜(π‘₯ +𝑒 𝑦)) ≀ ((πΉβ€˜π‘₯) +𝑒 (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ (𝐹 ∘ 𝐷) ∈ (∞Metβ€˜π‘‹))
 
Theorembdmetval 14403* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   (((𝐢:(𝑋 Γ— 𝑋)βŸΆβ„* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) = inf({(𝐴𝐢𝐡), 𝑅}, ℝ*, < ))
 
Theorembdxmet 14404* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
 
Theorembdmet 14405* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ+) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
 
Theorembdbl 14406* The standard bounded metric corresponding to 𝐢 generates the same balls as 𝐢 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≀ 𝑅)) β†’ (𝑃(ballβ€˜π·)𝑆) = (𝑃(ballβ€˜πΆ)𝑆))
 
Theorembdmopn 14407* The standard bounded metric corresponding to 𝐢 generates the same topology as 𝐢. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(π‘₯𝐢𝑦), 𝑅}, ℝ*, < ))    &   π½ = (MetOpenβ€˜πΆ)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) β†’ 𝐽 = (MetOpenβ€˜π·))
 
Theoremmopnex 14408* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (MetOpenβ€˜π·)    β‡’   (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ βˆƒπ‘‘ ∈ (Metβ€˜π‘‹)𝐽 = (MetOpenβ€˜π‘‘))
 
Theoremmetrest 14409 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
𝐷 = (𝐢 β†Ύ (π‘Œ Γ— π‘Œ))    &   π½ = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) = 𝐾)
 
Theoremxmetxp 14410* The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    β‡’   (πœ‘ β†’ 𝑃 ∈ (∞Metβ€˜(𝑋 Γ— π‘Œ)))
 
Theoremxmetxpbl 14411* The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐢 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   (πœ‘ β†’ 𝑅 ∈ ℝ*)    &   (πœ‘ β†’ 𝐢 ∈ (𝑋 Γ— π‘Œ))    β‡’   (πœ‘ β†’ (𝐢(ballβ€˜π‘ƒ)𝑅) = (((1st β€˜πΆ)(ballβ€˜π‘€)𝑅) Γ— ((2nd β€˜πΆ)(ballβ€˜π‘)𝑅)))
 
Theoremxmettxlem 14412* Lemma for xmettx 14413. (Contributed by Jim Kingdon, 15-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   π½ = (MetOpenβ€˜π‘€)    &   πΎ = (MetOpenβ€˜π‘)    &   πΏ = (MetOpenβ€˜π‘ƒ)    β‡’   (πœ‘ β†’ 𝐿 βŠ† (𝐽 Γ—t 𝐾))
 
Theoremxmettx 14413* The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑒 ∈ (𝑋 Γ— π‘Œ), 𝑣 ∈ (𝑋 Γ— π‘Œ) ↦ sup({((1st β€˜π‘’)𝑀(1st β€˜π‘£)), ((2nd β€˜π‘’)𝑁(2nd β€˜π‘£))}, ℝ*, < ))    &   (πœ‘ β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))    &   π½ = (MetOpenβ€˜π‘€)    &   πΎ = (MetOpenβ€˜π‘)    &   πΏ = (MetOpenβ€˜π‘ƒ)    β‡’   (πœ‘ β†’ 𝐿 = (𝐽 Γ—t 𝐾))
 
8.2.5  Continuity in metric spaces
 
Theoremmetcnp3 14414* Two ways to express that 𝐹 is continuous at 𝑃 for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ (𝐹 β€œ (𝑃(ballβ€˜πΆ)𝑧)) βŠ† ((πΉβ€˜π‘ƒ)(ballβ€˜π·)𝑦))))
 
Theoremmetcnp 14415* Two ways to say a mapping from metric 𝐢 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝑋 ((𝑃𝐢𝑀) < 𝑧 β†’ ((πΉβ€˜π‘ƒ)𝐷(πΉβ€˜π‘€)) < 𝑦))))
 
Theoremmetcnp2 14416* Two ways to say a mapping from metric 𝐢 to metric 𝐷 is continuous at point 𝑃. The distance arguments are swapped compared to metcnp 14415 (and Munkres' metcn 14417) for compatibility with df-lm 14093. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝑃 ∈ 𝑋) β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝑋 ((𝑀𝐢𝑃) < 𝑧 β†’ ((πΉβ€˜π‘€)𝐷(πΉβ€˜π‘ƒ)) < 𝑦))))
 
Theoremmetcn 14417* Two ways to say a mapping from metric 𝐢 to metric 𝐷 is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" 𝑦 there is a positive "delta" 𝑧 such that a distance less than delta in 𝐢 maps to a distance less than epsilon in 𝐷. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝑋 ((π‘₯𝐢𝑀) < 𝑧 β†’ ((πΉβ€˜π‘₯)𝐷(πΉβ€˜π‘€)) < 𝑦))))
 
Theoremmetcnpi 14418* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 14415. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ ℝ+)) β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘¦ ∈ 𝑋 ((𝑃𝐢𝑦) < π‘₯ β†’ ((πΉβ€˜π‘ƒ)𝐷(πΉβ€˜π‘¦)) < 𝐴))
 
Theoremmetcnpi2 14419* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 14416. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ ℝ+)) β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘¦ ∈ 𝑋 ((𝑦𝐢𝑃) < π‘₯ β†’ ((πΉβ€˜π‘¦)𝐷(πΉβ€˜π‘ƒ)) < 𝐴))
 
Theoremmetcnpi3 14420* Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 14419 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ∧ 𝐴 ∈ ℝ+)) β†’ βˆƒπ‘₯ ∈ ℝ+ βˆ€π‘¦ ∈ 𝑋 ((𝑦𝐢𝑃) ≀ π‘₯ β†’ ((πΉβ€˜π‘¦)𝐷(πΉβ€˜π‘ƒ)) ≀ 𝐴))
 
Theoremtxmetcnp 14421* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    &   πΏ = (MetOpenβ€˜πΈ)    β‡’   (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐸 ∈ (∞Metβ€˜π‘)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ π‘Œ)) β†’ (𝐹 ∈ (((𝐽 Γ—t 𝐾) CnP 𝐿)β€˜βŸ¨π΄, 𝐡⟩) ↔ (𝐹:(𝑋 Γ— π‘Œ)βŸΆπ‘ ∧ βˆ€π‘§ ∈ ℝ+ βˆƒπ‘€ ∈ ℝ+ βˆ€π‘’ ∈ 𝑋 βˆ€π‘£ ∈ π‘Œ (((𝐴𝐢𝑒) < 𝑀 ∧ (𝐡𝐷𝑣) < 𝑀) β†’ ((𝐴𝐹𝐡)𝐸(𝑒𝐹𝑣)) < 𝑧))))
 
Theoremtxmetcn 14422* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpenβ€˜πΆ)    &   πΎ = (MetOpenβ€˜π·)    &   πΏ = (MetOpenβ€˜πΈ)    β‡’   ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝐷 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐸 ∈ (∞Metβ€˜π‘)) β†’ (𝐹 ∈ ((𝐽 Γ—t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 Γ— π‘Œ)βŸΆπ‘ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ βˆ€π‘§ ∈ ℝ+ βˆƒπ‘€ ∈ ℝ+ βˆ€π‘’ ∈ 𝑋 βˆ€π‘£ ∈ π‘Œ (((π‘₯𝐢𝑒) < 𝑀 ∧ (𝑦𝐷𝑣) < 𝑀) β†’ ((π‘₯𝐹𝑦)𝐸(𝑒𝐹𝑣)) < 𝑧))))
 
Theoremmetcnpd 14423* Two ways to say a mapping from metric 𝐢 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.)
(πœ‘ β†’ 𝐽 = (MetOpenβ€˜πΆ))    &   (πœ‘ β†’ 𝐾 = (MetOpenβ€˜π·))    &   (πœ‘ β†’ 𝐢 ∈ (∞Metβ€˜π‘‹))    &   (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘Œ))    &   (πœ‘ β†’ 𝑃 ∈ 𝑋)    β‡’   (πœ‘ β†’ (𝐹 ∈ ((𝐽 CnP 𝐾)β€˜π‘ƒ) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ βˆ€π‘€ ∈ 𝑋 ((𝑃𝐢𝑀) < 𝑧 β†’ ((πΉβ€˜π‘ƒ)𝐷(πΉβ€˜π‘€)) < 𝑦))))
 
8.2.6  Topology on the reals
 
Theoremqtopbasss 14424* The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
𝑆 βŠ† ℝ*    &   ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ sup({π‘₯, 𝑦}, ℝ*, < ) ∈ 𝑆)    &   ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ inf({π‘₯, 𝑦}, ℝ*, < ) ∈ 𝑆)    β‡’   ((,) β€œ (𝑆 Γ— 𝑆)) ∈ TopBases
 
Theoremqtopbas 14425 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
((,) β€œ (β„š Γ— β„š)) ∈ TopBases
 
Theoremretopbas 14426 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
ran (,) ∈ TopBases
 
Theoremretop 14427 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
(topGenβ€˜ran (,)) ∈ Top
 
Theoremuniretop 14428 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
ℝ = βˆͺ (topGenβ€˜ran (,))
 
Theoremretopon 14429 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGenβ€˜ran (,)) ∈ (TopOnβ€˜β„)
 
Theoremretps 14430 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
𝐾 = {⟨(Baseβ€˜ndx), β„βŸ©, ⟨(TopSetβ€˜ndx), (topGenβ€˜ran (,))⟩}    β‡’   πΎ ∈ TopSp
 
Theoremiooretopg 14431 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴(,)𝐡) ∈ (topGenβ€˜ran (,)))
 
Theoremcnmetdval 14432 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐷 = (abs ∘ βˆ’ )    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴𝐷𝐡) = (absβ€˜(𝐴 βˆ’ 𝐡)))
 
Theoremcnmet 14433 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
(abs ∘ βˆ’ ) ∈ (Metβ€˜β„‚)
 
Theoremcnxmet 14434 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
 
Theoremcntoptopon 14435 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   π½ ∈ (TopOnβ€˜β„‚)
 
Theoremcntoptop 14436 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   π½ ∈ Top
 
Theoremcnbl0 14437 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐷 = (abs ∘ βˆ’ )    β‡’   (𝑅 ∈ ℝ* β†’ (β—‘abs β€œ (0[,)𝑅)) = (0(ballβ€˜π·)𝑅))
 
Theoremcnblcld 14438* Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐷 = (abs ∘ βˆ’ )    β‡’   (𝑅 ∈ ℝ* β†’ (β—‘abs β€œ (0[,]𝑅)) = {π‘₯ ∈ β„‚ ∣ (0𝐷π‘₯) ≀ 𝑅})
 
Theoremunicntopcntop 14439 The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
β„‚ = βˆͺ (MetOpenβ€˜(abs ∘ βˆ’ ))
 
Theoremcnopncntop 14440 The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.)
β„‚ ∈ (MetOpenβ€˜(abs ∘ βˆ’ ))
 
Theoremreopnap 14441* The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
(𝐴 ∈ ℝ β†’ {𝑀 ∈ ℝ ∣ 𝑀 # 𝐴} ∈ (topGenβ€˜ran (,)))
 
Theoremremetdval 14442 Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴𝐷𝐡) = (absβ€˜(𝐴 βˆ’ 𝐡)))
 
Theoremremet 14443 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   π· ∈ (Metβ€˜β„)
 
Theoremrexmet 14444 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   π· ∈ (∞Metβ€˜β„)
 
Theorembl2ioo 14445 A ball in terms of an open interval of reals. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴(ballβ€˜π·)𝐡) = ((𝐴 βˆ’ 𝐡)(,)(𝐴 + 𝐡)))
 
Theoremioo2bl 14446 An open interval of reals in terms of a ball. (Contributed by NM, 18-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴(,)𝐡) = (((𝐴 + 𝐡) / 2)(ballβ€˜π·)((𝐡 βˆ’ 𝐴) / 2)))
 
Theoremioo2blex 14447 An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   ((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴(,)𝐡) ∈ ran (ballβ€˜π·))
 
Theoremblssioo 14448 The balls of the standard real metric space are included in the open real intervals. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    β‡’   ran (ballβ€˜π·) βŠ† ran (,)
 
Theoremtgioo 14449 The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))    &   π½ = (MetOpenβ€˜π·)    β‡’   (topGenβ€˜ran (,)) = 𝐽
 
Theoremtgqioo 14450 The topology generated by open intervals of reals with rational endpoints is the same as the open sets of the standard metric space on the reals. In particular, this proves that the standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝑄 = (topGenβ€˜((,) β€œ (β„š Γ— β„š)))    β‡’   (topGenβ€˜ran (,)) = 𝑄
 
Theoremresubmet 14451 The subspace topology induced by a subset of the reals. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Aug-2014.)
𝑅 = (topGenβ€˜ran (,))    &   π½ = (MetOpenβ€˜((abs ∘ βˆ’ ) β†Ύ (𝐴 Γ— 𝐴)))    β‡’   (𝐴 βŠ† ℝ β†’ 𝐽 = (𝑅 β†Ύt 𝐴))
 
Theoremtgioo2cntop 14452 The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (topGenβ€˜ran (,)) = (𝐽 β†Ύt ℝ)
 
Theoremrerestcntop 14453 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   π‘… = (topGenβ€˜ran (,))    β‡’   (𝐴 βŠ† ℝ β†’ (𝐽 β†Ύt 𝐴) = (𝑅 β†Ύt 𝐴))
 
Theoremaddcncntoplem 14454* Lemma for addcncntop 14455, subcncntop 14456, and mulcncntop 14457. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &    + :(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)
 
Theoremaddcncntop 14455 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 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’    + ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremsubcncntop 14456 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 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’    βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremmulcncntop 14457 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 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’    Β· ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)
 
Theoremdivcnap 14458* Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   πΎ = (𝐽 β†Ύt {π‘₯ ∈ β„‚ ∣ π‘₯ # 0})    β‡’   (𝑦 ∈ β„‚, 𝑧 ∈ {π‘₯ ∈ β„‚ ∣ π‘₯ # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽)
 
Theoremfsumcncntop 14459* 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 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   (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))
 
8.2.7  Topological definitions using the reals
 
Syntaxccncf 14460 Extend class notation to include the operation which returns a class of continuous complex functions.
class –cnβ†’
 
Definitiondf-cncf 14461* 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 β†‘π‘š 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)})
 
Theoremcncfval 14462* 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 β†‘π‘š 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 14463* 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 14464* Version of elcncf 14463 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 14465 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴–cn→𝐡) β†’ 𝐴 βŠ† β„‚)
 
Theoremcncfrss2 14466 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴–cn→𝐡) β†’ 𝐡 βŠ† β„‚)
 
Theoremcncff 14467 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 14468* 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 14469* 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 14470* 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 14471 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 14472 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 14473 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 14474 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 14475 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)
 
Theoremrecncf 14476 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
β„œ ∈ (ℂ–cn→ℝ)
 
Theoremimcncf 14477 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
β„‘ ∈ (ℂ–cn→ℝ)
 
Theoremcjcncf 14478 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
βˆ— ∈ (ℂ–cnβ†’β„‚)
 
Theoremmulc1cncf 14479* 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„‚))
 
Theoremdivccncfap 14480* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ / 𝐴))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 # 0) β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremcncfco 14481 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))
 
Theoremcncfmet 14482 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))
 
Theoremcncfcncntop 14483 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   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))
 
Theoremcncfcn1cntop 14484 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   (ℂ–cnβ†’β„‚) = (𝐽 Cn 𝐽)
 
Theoremcncfmptc 14485* 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 14486* 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 14487* Composition of continuous functions. –cnβ†’ analogue of cnmpt11f 14187. (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„‚))
 
Theoremcncfmpt2fcntop 14488* Composition of continuous functions. –cnβ†’ analogue of cnmpt12f 14189. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))    &   (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„‚))
 
Theoremaddccncf 14489* Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝐹 = (π‘₯ ∈ β„‚ ↦ (π‘₯ + 𝐴))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ (ℂ–cnβ†’β„‚))
 
Theoremcdivcncfap 14490* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.)
𝐹 = (π‘₯ ∈ {𝑦 ∈ β„‚ ∣ 𝑦 # 0} ↦ (𝐴 / π‘₯))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ∈ ({𝑦 ∈ β„‚ ∣ 𝑦 # 0}–cnβ†’β„‚))
 
Theoremnegcncf 14491* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ -π‘₯)    β‡’   (𝐴 βŠ† β„‚ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))
 
Theoremnegfcncf 14492* 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„‚))
 
Theoremmulcncflem 14493* Lemma for mulcncf 14494. (Contributed by Jim Kingdon, 29-May-2023.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ 𝑉 ∈ 𝑋)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐹 ∈ ℝ+)    &   (πœ‘ β†’ 𝐺 ∈ ℝ+)    &   (πœ‘ β†’ 𝑆 ∈ ℝ+)    &   (πœ‘ β†’ 𝑇 ∈ ℝ+)    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑆 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ 𝐴)β€˜π‘‰))) < 𝐹))    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑇 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ 𝐡)β€˜π‘‰))) < 𝐺))    &   (πœ‘ β†’ βˆ€π‘’ ∈ 𝑋 (((absβ€˜(⦋𝑒 / π‘₯⦌𝐴 βˆ’ ⦋𝑉 / π‘₯⦌𝐴)) < 𝐹 ∧ (absβ€˜(⦋𝑒 / π‘₯⦌𝐡 βˆ’ ⦋𝑉 / π‘₯⦌𝐡)) < 𝐺) β†’ (absβ€˜((⦋𝑒 / π‘₯⦌𝐴 Β· ⦋𝑒 / π‘₯⦌𝐡) βˆ’ (⦋𝑉 / π‘₯⦌𝐴 Β· ⦋𝑉 / π‘₯⦌𝐡))) < 𝐸))    β‡’   (πœ‘ β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘’ ∈ 𝑋 ((absβ€˜(𝑒 βˆ’ 𝑉)) < 𝑑 β†’ (absβ€˜(((π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡))β€˜π‘’) βˆ’ ((π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡))β€˜π‘‰))) < 𝐸))
 
Theoremmulcncf 14494* The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cnβ†’β„‚))    &   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ 𝐡) ∈ (𝑋–cnβ†’β„‚))    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝑋 ↦ (𝐴 Β· 𝐡)) ∈ (𝑋–cnβ†’β„‚))
 
Theoremexpcncf 14495* The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
(𝑁 ∈ β„•0 β†’ (π‘₯ ∈ β„‚ ↦ (π‘₯↑𝑁)) ∈ (ℂ–cnβ†’β„‚))
 
Theoremcnrehmeocntop 14496* The canonical bijection from (ℝ Γ— ℝ) to β„‚ described in cnref1o 9669 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.)
𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (π‘₯ + (i Β· 𝑦)))    &   π½ = (topGenβ€˜ran (,))    &   πΎ = (MetOpenβ€˜(abs ∘ βˆ’ ))    β‡’   πΉ ∈ ((𝐽 Γ—t 𝐽)Homeo𝐾)
 
Theoremcnopnap 14497* The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.)
(𝐴 ∈ β„‚ β†’ {𝑀 ∈ β„‚ ∣ 𝑀 # 𝐴} ∈ (MetOpenβ€˜(abs ∘ βˆ’ )))
 
PART 9  BASIC REAL AND COMPLEX ANALYSIS
 
9.0.1  Dedekind cuts
 
Theoremdedekindeulemuub 14498* Lemma for dedekindeu 14504. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    &   (πœ‘ β†’ 𝐴 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝐴)
 
Theoremdedekindeulemub 14499* Lemma for dedekindeu 14504. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐿 𝑦 < π‘₯)
 
Theoremdedekindeulemloc 14500* Lemma for dedekindeu 14504. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.)
(πœ‘ β†’ 𝐿 βŠ† ℝ)    &   (πœ‘ β†’ π‘ˆ βŠ† ℝ)    &   (πœ‘ β†’ βˆƒπ‘ž ∈ ℝ π‘ž ∈ 𝐿)    &   (πœ‘ β†’ βˆƒπ‘Ÿ ∈ ℝ π‘Ÿ ∈ π‘ˆ)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ (π‘ž ∈ 𝐿 ↔ βˆƒπ‘Ÿ ∈ 𝐿 π‘ž < π‘Ÿ))    &   (πœ‘ β†’ βˆ€π‘Ÿ ∈ ℝ (π‘Ÿ ∈ π‘ˆ ↔ βˆƒπ‘ž ∈ π‘ˆ π‘ž < π‘Ÿ))    &   (πœ‘ β†’ (𝐿 ∩ π‘ˆ) = βˆ…)    &   (πœ‘ β†’ βˆ€π‘ž ∈ ℝ βˆ€π‘Ÿ ∈ ℝ (π‘ž < π‘Ÿ β†’ (π‘ž ∈ 𝐿 ∨ π‘Ÿ ∈ π‘ˆ)))    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ ℝ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ (βˆƒπ‘§ ∈ 𝐿 π‘₯ < 𝑧 ∨ βˆ€π‘§ ∈ 𝐿 𝑧 < 𝑦)))
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