P. 248 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xrsmopn 24701	The metric on the extended reals generates a topology, but this does not match the order topology on ℝ*; for example {+∞} is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (ordTop‘ ≤ ) ⊆ 𝐽
Theorem	zcld 24702	The integers are a closed set in the topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (topGen‘ran (,))    ⇒   ⊢ ℤ ∈ (Clsd‘𝐽)
Theorem	recld2 24703	The real numbers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ℝ ∈ (Clsd‘𝐽)
Theorem	zcld2 24704	The integers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ℤ ∈ (Clsd‘𝐽)
Theorem	zdis 24705	The integers are a discrete set in the topology on ℂ. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐽 ↾t ℤ) = 𝒫 ℤ
Theorem	sszcld 24706	Every subset of the integers are closed in the topology on ℂ. (Contributed by Mario Carneiro, 6-Jul-2017.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽))
Theorem	reperflem 24707*	A subset of the real numbers that is closed under addition with real numbers is perfect. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ 𝑆)    &   ⊢ 𝑆 ⊆ ℂ    ⇒   ⊢ (𝐽 ↾t 𝑆) ∈ Perf
Theorem	reperf 24708	The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐽 ↾t ℝ) ∈ Perf
Theorem	cnperf 24709	The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Perf
Theorem	iccntr 24710	The interior of a closed interval in the standard topology on ℝ is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
Theorem	icccmplem1 24711*	Lemma for icccmp 24714. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵))
Theorem	icccmplem2 24712*	Lemma for icccmp 24714. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    &   ⊢ (𝜑 → 𝑉 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐺(ball‘𝐷)𝐶) ⊆ 𝑉)    &   ⊢ 𝐺 = sup(𝑆, ℝ, < )    &   ⊢ 𝑅 = if((𝐺 + (𝐶 / 2)) ≤ 𝐵, (𝐺 + (𝐶 / 2)), 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑆)
Theorem	icccmplem3 24713*	Lemma for icccmp 24714. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧}    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑈 ⊆ 𝐽)    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑆)
Theorem	icccmp 24714	A closed interval in ℝ is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
Theorem	reconnlem1 24715	Lemma for reconn 24717. Connectedness in the reals-easy direction. (Contributed by Jeff Hankins, 13-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (((𝐴 ⊆ ℝ ∧ ((topGen‘ran (,)) ↾t 𝐴) ∈ Conn) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴)
Theorem	reconnlem2 24716*	Lemma for reconn 24717. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑈 ∈ (topGen‘ran (,)))    &   ⊢ (𝜑 → 𝑉 ∈ (topGen‘ran (,)))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∩ 𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑉 ∩ 𝐴))    &   ⊢ (𝜑 → (𝑈 ∩ 𝑉) ⊆ (ℝ ∖ 𝐴))    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ 𝑆 = sup((𝑈 ∩ (𝐵[,]𝐶)), ℝ, < )    ⇒   ⊢ (𝜑 → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉))
Theorem	reconn 24717*	A subset of the reals is connected iff it has the interval property. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
⊢ (𝐴 ⊆ ℝ → (((topGen‘ran (,)) ↾t 𝐴) ∈ Conn ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴))
Theorem	retopconn 24718	Corollary of reconn 24717. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ (topGen‘ran (,)) ∈ Conn
Theorem	iccconn 24719	A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
Theorem	opnreen 24720	Every nonempty open set is uncountable. (Contributed by Mario Carneiro, 26-Jul-2014.) (Revised by Mario Carneiro, 20-Feb-2015.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → 𝐴 ≈ 𝒫 ℕ)
Theorem	rectbntr0 24721	A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) = ∅)
Theorem	xrge0gsumle 24722	A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∩ Fin))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ≤ (𝐺 Σg (𝐹 ↾ 𝐵)))
Theorem	xrge0tsms 24723*	Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Proof shortened by AV, 26-Jul-2019.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))), ℝ*, < )    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝑆})
Theorem	xrge0tsms2 24724	Any finite or infinite sum in the nonnegative extended reals is convergent. This is a rather unique property of the set [0, +∞]; a similar theorem is not true for ℝ* or ℝ or [0, +∞). It is true for ℕ0 ∪ {+∞}, however, or more generally any additive submonoid of [0, +∞) with +∞ adjoined. (Contributed by Mario Carneiro, 13-Sep-2015.)
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o)
Theorem	metdcnlem 24725	The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐶 = (dist‘ℝ*𝑠)    &   ⊢ 𝐾 = (MetOpen‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐴𝐷𝑌) < (𝑅 / 2))    &   ⊢ (𝜑 → (𝐵𝐷𝑍) < (𝑅 / 2))    ⇒   ⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) < 𝑅)
Theorem	xmetdcn2 24726	The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 24727 we use the metric topology instead of the order topology on ℝ*, which makes the theorem a bit stronger. Since +∞ is an isolated point in the metric topology, this is saying that for any points 𝐴, 𝐵 which are an infinite distance apart, there is a product neighborhood around ⟨𝐴, 𝐵⟩ such that 𝑑(𝑎, 𝑏) = +∞ for any 𝑎 near 𝐴 and 𝑏 near 𝐵, i.e., the distance function is locally constant +∞. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐶 = (dist‘ℝ*𝑠)    &   ⊢ 𝐾 = (MetOpen‘𝐶)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
Theorem	xmetdcn 24727	The metric function of an extended metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (ordTop‘ ≤ )    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
Theorem	metdcn2 24728	The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
Theorem	metdcn 24729	The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
Theorem	msdcn 24730	The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    &   ⊢ 𝐽 = (TopOpen‘𝑀)    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ ((𝐽 ×t 𝐽) Cn 𝐾))
Theorem	cnmpt1ds 24731*	Continuity of the metric function; analogue of cnmpt12f 23553 which cannot be used directly because 𝐷 is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑅 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐺 ∈ MetSp)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐷𝐵)) ∈ (𝐾 Cn 𝑅))
Theorem	cnmpt2ds 24732*	Continuity of the metric function; analogue of cnmpt22f 23562 which cannot be used directly because 𝐷 is not necessarily a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐷 = (dist‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝑅 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐺 ∈ MetSp)    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐷𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅))
Theorem	nmcn 24733	The norm of a normed group is a continuous function. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾))
Theorem	ngnmcncn 24734	The norm of a normed group is a continuous function to ℂ. (Contributed by NM, 12-Aug-2007.) (Revised by AV, 17-Oct-2021.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ 𝐽 = (TopOpen‘𝐺)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾))
Theorem	abscn 24735	The absolute value function on complex numbers is continuous. (Contributed by NM, 22-Aug-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (topGen‘ran (,))    ⇒   ⊢ abs ∈ (𝐽 Cn 𝐾)
Theorem	metdsval 24736*	Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Revised by AV, 30-Sep-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
Theorem	metdsf 24737*	The distance from a point to a set is a nonnegative extended real number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
Theorem	metdsge 24738*	The distance from the point 𝐴 to the set 𝑆 is greater than 𝑅 iff the 𝑅-ball around 𝐴 misses 𝑆. (Contributed by Mario Carneiro, 4-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))
Theorem	metds0 24739*	If a point is in a set, its distance to the set is zero. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0)
Theorem	metdstri 24740*	A generalization of the triangle inequality to the point-set distance function. Under the usual notation where the same symbol 𝑑 denotes the point-point and point-set distance functions, this theorem would be written 𝑑(𝑎, 𝑆) ≤ 𝑑(𝑎, 𝑏) + 𝑑(𝑏, 𝑆). (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵)))
Theorem	metdsle 24741*	The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵))
Theorem	metdsre 24742*	The distance from a point to a nonempty set in a proper metric space is a real number. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    ⇒   ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝐹:𝑋⟶ℝ)
Theorem	metdseq0 24743*	The distance from a point to a set is zero iff the point is in the closure set. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆)))
Theorem	metdscnlem 24744*	Lemma for metdscn 24745. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐶 = (dist‘ℝ*𝑠)    &   ⊢ 𝐾 = (MetOpen‘𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅)
Theorem	metdscn 24745*	The function 𝐹 which gives the distance from a point to a set is a continuous function into the metric topology of the extended reals. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐶 = (dist‘ℝ*𝑠)    &   ⊢ 𝐾 = (MetOpen‘𝐶)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	metdscn2 24746*	The function 𝐹 which gives the distance from a point to a nonempty set in a metric space is a continuous function into the topology of the complex numbers. (Contributed by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	metnrmlem1a 24747*	Lemma for metnrm 24751. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑇) → (0 < (𝐹‘𝐴) ∧ if(1 ≤ (𝐹‘𝐴), 1, (𝐹‘𝐴)) ∈ ℝ+))
Theorem	metnrmlem1 24748*	Lemma for metnrm 24751. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    ⇒   ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐴𝐷𝐵))
Theorem	metnrmlem2 24749*	Lemma for metnrm 24751. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))    ⇒   ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈))
Theorem	metnrmlem3 24750*	Lemma for metnrm 24751. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))    &   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))    &   ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   ⊢ 𝑉 = ∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2))    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
Theorem	metnrm 24751	A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm)
Theorem	metreg 24752	A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
Theorem	addcnlem 24753*	Lemma for addcn 24754, subcn 24755, and mulcn 24756. (Contributed by Mario Carneiro, 5-May-2014.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ + :(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	addcn 24754	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 = (TopOpen‘ℂfld)    ⇒   ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	subcn 24755	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 = (TopOpen‘ℂfld)    ⇒   ⊢ − ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	mulcn 24756	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.) Usage of this theorem is discouraged because it depends on ax-mulf 11148. Use mpomulcn 24758 instead. (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	divcnOLD 24757	Obsolete version of divcn 24759 as of 6-Apr-2025. (Contributed by Mario Carneiro, 12-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0}))    ⇒   ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
Theorem	mpomulcn 24758*	Complex number multiplication is a continuous function. Version of mulcn 24756 using maps-to notation, which does not require ax-mulf 11148. (Contributed by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	divcn 24759	Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) Avoid ax-mulf 11148. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0}))    ⇒   ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
Theorem	cnfldtgp 24760	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.)
⊢ ℂfld ∈ TopGrp
Theorem	fsumcn 24761*	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	fsum2cn 24762*	Version of fsumcn 24761 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))
Theorem	expcn 24763*	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 11148. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
Theorem	divccn 24764*	Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.) Avoid ax-mulf 11148. (Revised by GG, 16-Mar-2025.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
Theorem	expcnOLD 24765*	Obsolete version of expcn 24763 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 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
Theorem	divccnOLD 24766*	Obsolete version of divccn 24764 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
Theorem	sqcn 24767*	The square function on complex numbers is continuous. (Contributed by NM, 13-Jun-2007.) (Proof shortened by Mario Carneiro, 5-May-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑2)) ∈ (𝐽 Cn 𝐽)
12.4.11  Topological definitions using the reals
Syntax	cii 24768	Extend class notation with the unit interval.
class II
Syntax	ccncf 24769	Extend class notation to include the operation which returns a class of continuous complex functions.
class –cn→
Definition	df-ii 24770	Define the unit interval with the Euclidean topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
Definition	df-cncf 24771*	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	iitopon 24772	The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ II ∈ (TopOn‘(0[,]1))
Theorem	iitop 24773	The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II ∈ Top
Theorem	iiuni 24774	The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
⊢ (0[,]1) = ∪ II
Theorem	dfii2 24775	Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
Theorem	dfii3 24776	Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ II = (𝐽 ↾t (0[,]1))
Theorem	dfii4 24777	Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐼 = (ℂfld ↾s (0[,]1))    ⇒   ⊢ II = (TopOpen‘𝐼)
Theorem	dfii5 24778	The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
Theorem	iicmp 24779	The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
⊢ II ∈ Comp
Theorem	iiconn 24780	The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ II ∈ Conn
Theorem	cncfval 24781*	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 24782*	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 24783*	Version of elcncf 24782 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 24784	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
Theorem	cncfrss2 24785	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
Theorem	cncff 24786	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 24787*	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 24788*	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 24789*	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))
Theorem	rescncf 24790	A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐶 ⊆ 𝐴 → (𝐹 ∈ (𝐴–cn→𝐵) → (𝐹 ↾ 𝐶) ∈ (𝐶–cn→𝐵)))
Theorem	cncfcdm 24791	Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
⊢ ((𝐶 ⊆ ℂ ∧ 𝐹 ∈ (𝐴–cn→𝐵)) → (𝐹 ∈ (𝐴–cn→𝐶) ↔ 𝐹:𝐴⟶𝐶))
Theorem	cncfss 24792	The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶))
Theorem	climcncf 24793	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝐴)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐷)    &   ⊢ (𝜑 → 𝐷 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝ (𝐹‘𝐷))
Theorem	abscncf 24794	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ abs ∈ (ℂ–cn→ℝ)
Theorem	recncf 24795	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℜ ∈ (ℂ–cn→ℝ)
Theorem	imcncf 24796	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℑ ∈ (ℂ–cn→ℝ)
Theorem	cjcncf 24797	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ∗ ∈ (ℂ–cn→ℂ)
Theorem	mulc1cncf 24798*	Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 · 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	divccncf 24799*	Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	cncfco 24800	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.)
⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐵–cn→𝐶))    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ∈ (𝐴–cn→𝐶))