P. 149 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	setsmstsetg 14801	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → (MetOpen‘𝐷) ∈ 𝑊)    ⇒   ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))
Theorem	mopni 14802*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
Theorem	mopni2 14803*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)
Theorem	mopni3 14804*	An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))
Theorem	blssopn 14805	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝐽)
Theorem	unimopn 14806	The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)
Theorem	mopnin 14807	The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
Theorem	mopn0 14808	The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∅ ∈ 𝐽)
Theorem	rnblopn 14809	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷)) → 𝐵 ∈ 𝐽)
Theorem	blopn 14810	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)
Theorem	neibl 14811*	The neighborhoods around a point 𝑃 of a metric space are those subsets containing a ball around 𝑃. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))
Theorem	blnei 14812	A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))
Theorem	blsscls2 14813*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝑆 = {𝑧 ∈ 𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑇 ∈ ℝ* ∧ 𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))
Theorem	metss 14814*	Two ways of saying that metric 𝐷 generates a finer topology than metric 𝐶. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
Theorem	metequiv 14815*	Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 = 𝐾 ↔ ∀𝑥 ∈ 𝑋 (∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))
Theorem	metequiv2 14816*	If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ (𝑠 ≤ 𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))
Theorem	metss2lem 14817*	Lemma for metss2 14818. (Contributed by Mario Carneiro, 14-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))
Theorem	metss2 14818*	If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 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 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))    ⇒   ⊢ (𝜑 → 𝐽 ⊆ 𝐾)
Theorem	comet 14819*	The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐹:(0[,]+∞)⟶ℝ*)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹‘𝑥) +𝑒 (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐷) ∈ (∞Met‘𝑋))
Theorem	bdmetval 14820*	Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ* ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ))
Theorem	bdxmet 14821*	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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
Theorem	bdmet 14822*	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 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
Theorem	bdbl 14823*	The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ* ∧ 𝑆 ≤ 𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
Theorem	bdmopn 14824*	The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
Theorem	mopnex 14825*	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 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑))
Theorem	metrest 14826	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 = (𝐶 ↾ (𝑌 × 𝑌))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)
Theorem	xmetxp 14827*	The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    ⇒   ⊢ (𝜑 → 𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
Theorem	xmetxpbl 14828*	The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐶 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)
⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝑋 × 𝑌))    ⇒   ⊢ (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1st ‘𝐶)(ball‘𝑀)𝑅) × ((2nd ‘𝐶)(ball‘𝑁)𝑅)))
Theorem	xmettxlem 14829*	Lemma for xmettx 14830. (Contributed by Jim Kingdon, 15-Oct-2023.)
⊢ 𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ 𝐽 = (MetOpen‘𝑀)    &   ⊢ 𝐾 = (MetOpen‘𝑁)    &   ⊢ 𝐿 = (MetOpen‘𝑃)    ⇒   ⊢ (𝜑 → 𝐿 ⊆ (𝐽 ×t 𝐾))
Theorem	xmettx 14830*	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 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st ‘𝑢)𝑀(1st ‘𝑣)), ((2nd ‘𝑢)𝑁(2nd ‘𝑣))}, ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌))    &   ⊢ 𝐽 = (MetOpen‘𝑀)    &   ⊢ 𝐾 = (MetOpen‘𝑁)    &   ⊢ 𝐿 = (MetOpen‘𝑃)    ⇒   ⊢ (𝜑 → 𝐿 = (𝐽 ×t 𝐾))
9.2.5  Continuity in metric spaces
Theorem	metcnp3 14831*	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 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
Theorem	metcnp 14832*	Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
Theorem	metcnp2 14833*	Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. The distance arguments are swapped compared to metcnp 14832 (and Munkres' metcn 14834) for compatibility with df-lm 14510. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑤𝐶𝑃) < 𝑧 → ((𝐹‘𝑤)𝐷(𝐹‘𝑃)) < 𝑦))))
Theorem	metcn 14834*	Two ways to say a mapping from metric 𝐶 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 𝐶 maps to a distance less than epsilon in 𝐷. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝐹‘𝑥)𝐷(𝐹‘𝑤)) < 𝑦))))
Theorem	metcnpi 14835*	Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 14832. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹‘𝑃)𝐷(𝐹‘𝑦)) < 𝐴))
Theorem	metcnpi2 14836*	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 14833. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))
Theorem	metcnpi3 14837*	Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 14836 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
Theorem	txmetcnp 14838*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ 𝐿 = (MetOpen‘𝐸)    ⇒   ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
Theorem	txmetcn 14839*	Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    &   ⊢ 𝐿 = (MetOpen‘𝐸)    ⇒   ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ∀𝑧 ∈ ℝ+ ∃𝑤 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
Theorem	metcnpd 14840*	Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.)
⊢ (𝜑 → 𝐽 = (MetOpen‘𝐶))    &   ⊢ (𝜑 → 𝐾 = (MetOpen‘𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
9.2.6  Topology on the reals
Theorem	qtopbasss 14841*	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 ⊆ ℝ*    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → sup({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)    &   ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → inf({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)    ⇒   ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases
Theorem	qtopbas 14842	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
Theorem	retopbas 14843	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
Theorem	retop 14844	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
⊢ (topGen‘ran (,)) ∈ Top
Theorem	uniretop 14845	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
⊢ ℝ = ∪ (topGen‘ran (,))
Theorem	retopon 14846	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
Theorem	retps 14847	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	iooretopg 14848	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* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
Theorem	cnmetdval 14849	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 ∘ − )    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
Theorem	cnmet 14850	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 ∘ − ) ∈ (Met‘ℂ)
Theorem	cnxmet 14851	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
Theorem	cntoptopon 14852	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cntoptop 14853	The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ 𝐽 ∈ Top
Theorem	cnbl0 14854	Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))
Theorem	cnblcld 14855*	Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅})
Theorem	cnfldms 14856	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ MetSp
Theorem	cnfldxms 14857	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ ∞MetSp
Theorem	cnfldtps 14858	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ TopSp
Theorem	cnfldtopn 14859	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
Theorem	cnfldtopon 14860	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cnfldtop 14861	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	unicntopcntop 14862	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.)
⊢ ℂ = ∪ (MetOpen‘(abs ∘ − ))
Theorem	unicntop 14863	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ = ∪ (TopOpen‘ℂfld)
Theorem	cnopncntop 14864	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.)
⊢ ℂ ∈ (MetOpen‘(abs ∘ − ))
Theorem	cnopn 14865	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ ∈ (TopOpen‘ℂfld)
Theorem	reopnap 14866*	The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.)
⊢ (𝐴 ∈ ℝ → {𝑤 ∈ ℝ ∣ 𝑤 # 𝐴} ∈ (topGen‘ran (,)))
Theorem	remetdval 14867	Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
Theorem	remet 14868	The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐷 ∈ (Met‘ℝ)
Theorem	rexmet 14869	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ)
Theorem	bl2ioo 14870	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 ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(ball‘𝐷)𝐵) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵)))
Theorem	ioo2bl 14871	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 ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) = (((𝐴 + 𝐵) / 2)(ball‘𝐷)((𝐵 − 𝐴) / 2)))
Theorem	ioo2blex 14872	An open interval of reals in terms of a ball. (Contributed by Mario Carneiro, 14-Nov-2013.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴(,)𝐵) ∈ ran (ball‘𝐷))
Theorem	blssioo 14873	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 ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ran (ball‘𝐷) ⊆ ran (,)
Theorem	tgioo 14874	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 ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (topGen‘ran (,)) = 𝐽
Theorem	tgqioo 14875	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 = (topGen‘((,) “ (ℚ × ℚ)))    ⇒   ⊢ (topGen‘ran (,)) = 𝑄
Theorem	resubmet 14876	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 = (topGen‘ran (,))    &   ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))    ⇒   ⊢ (𝐴 ⊆ ℝ → 𝐽 = (𝑅 ↾t 𝐴))
Theorem	tgioo2cntop 14877	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 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ (topGen‘ran (,)) = (𝐽 ↾t ℝ)
Theorem	rerestcntop 14878	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 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝑅 = (topGen‘ran (,))    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴))
Theorem	tgioo2 14879	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (topGen‘ran (,)) = (𝐽 ↾t ℝ)
Theorem	rerest 14880	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑅 = (topGen‘ran (,))    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴))
Theorem	addcncntoplem 14881*	Lemma for addcncntop 14882, subcncntop 14883, and mulcncntop 14884. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.)
⊢ 𝐽 = (MetOpen‘(abs ∘ − ))    &   ⊢ + :(ℂ × ℂ)⟶ℂ    &   ⊢ ((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℂ ∧ 𝑐 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝑏)) < 𝑦 ∧ (abs‘(𝑣 − 𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎))    ⇒   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	addcncntop 14882	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 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	subcncntop 14883	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 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	mulcncntop 14884	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 = (MetOpen‘(abs ∘ − ))    ⇒   ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	divcnap 14885*	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 = (MetOpen‘(abs ∘ − ))    &   ⊢ 𝐾 = (𝐽 ↾t {𝑥 ∈ ℂ ∣ 𝑥 # 0})    ⇒   ⊢ (𝑦 ∈ ℂ, 𝑧 ∈ {𝑥 ∈ ℂ ∣ 𝑥 # 0} ↦ (𝑦 / 𝑧)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
Theorem	mpomulcn 14886*	Complex number multiplication is a continuous function. (Contributed by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	fsumcncntop 14887*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐾 = (MetOpen‘(abs ∘ − ))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	fsumcn 14888*	A finite sum of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by NM, 8-Aug-2007.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	expcn 14889*	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 8019. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
9.2.7  Topological definitions using the reals
Syntax	ccncf 14890	Extend class notation to include the operation which returns a class of continuous complex functions.
class –cn→
Definition	df-cncf 14891*	Define the operation whose value is a class of continuous complex functions. (Contributed by Paul Chapman, 11-Oct-2007.)
⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑𝑚 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)})
Theorem	cncfval 14892*	The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
Theorem	elcncf 14893*	Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))))
Theorem	elcncf2 14894*	Version of elcncf 14893 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦))))
Theorem	cncfrss 14895	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
Theorem	cncfrss2 14896	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
Theorem	cncff 14897	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→𝐵) → 𝐹:𝐴⟶𝐵)
Theorem	cncfi 14898*	Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅))
Theorem	elcncf1di 14899*	Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))    ⇒   ⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵)))
Theorem	elcncf1ii 14900*	Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
⊢ 𝐹:𝐴⟶𝐵    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))