P. 240 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rectbntr0 23901	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 23902	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 23903*	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 23904	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 23905	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 23906	The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 23907 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 23907	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 23908	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 23909	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 23910	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 23911*	Continuity of the metric function; analogue of cnmpt12f 22725 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 23912*	Continuity of the metric function; analogue of cnmpt22f 22734 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 23913	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 23914	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 23915	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 23916*	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 23917*	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 23918*	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 23919*	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 23920*	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 23921*	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 23922*	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 23923*	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 23924*	Lemma for metdscn 23925. (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 23925*	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 23926*	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 23927*	Lemma for metnrm 23931. (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 23928*	Lemma for metnrm 23931. (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 23929*	Lemma for metnrm 23931. (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 23930*	Lemma for metnrm 23931. (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 23931	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 23932	A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
Theorem	addcnlem 23933*	Lemma for addcn 23934, subcn 23935, and mulcn 23936. (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 23934	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 23935	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 23936	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 = (TopOpen‘ℂfld)    ⇒   ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Theorem	divcn 23937	Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0}))    ⇒   ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
Theorem	cnfldtgp 23938	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 23939*	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 23940*	Version of fsumcn 23939 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 23941*	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.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))
Theorem	divccn 23942*	Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽))
Theorem	sqcn 23943*	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 23944	Extend class notation with the unit interval.
class II
Syntax	ccncf 23945	Extend class notation to include the operation which returns a class of continuous complex functions.
class –cn→
Definition	df-ii 23946	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 23947*	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 23948	The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ II ∈ (TopOn‘(0[,]1))
Theorem	iitop 23949	The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II ∈ Top
Theorem	iiuni 23950	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 23951	Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
Theorem	dfii3 23952	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 23953	Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐼 = (ℂfld ↾s (0[,]1))    ⇒   ⊢ II = (TopOpen‘𝐼)
Theorem	dfii5 23954	The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
Theorem	iicmp 23955	The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
⊢ II ∈ Comp
Theorem	iiconn 23956	The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ II ∈ Conn
Theorem	cncfval 23957*	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 23958*	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 23959*	Version of elcncf 23958 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 23960	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ)
Theorem	cncfrss2 23961	Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)
Theorem	cncff 23962	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 23963*	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 23964*	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 23965*	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 23966	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	cncffvrn 23967	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 23968	The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶))
Theorem	climcncf 23969	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 23970	Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ abs ∈ (ℂ–cn→ℝ)
Theorem	recncf 23971	Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℜ ∈ (ℂ–cn→ℝ)
Theorem	imcncf 23972	Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ℑ ∈ (ℂ–cn→ℝ)
Theorem	cjcncf 23973	Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
⊢ ∗ ∈ (ℂ–cn→ℂ)
Theorem	mulc1cncf 23974*	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 23975*	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 23976	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))
Theorem	cncfcompt2 23977*	Composition of continuous functions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑅) ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑆) ∈ (𝐶–cn→𝐸))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑇) ∈ (𝐴–cn→𝐸))
Theorem	cncfmet 23978	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ 𝐶 = ((abs ∘ − ) ↾ (𝐴 × 𝐴))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (MetOpen‘𝐶)    &   ⊢ 𝐾 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐽 Cn 𝐾))
Theorem	cncfcn 23979	Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐴)    &   ⊢ 𝐿 = (𝐽 ↾t 𝐵)    ⇒   ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = (𝐾 Cn 𝐿))
Theorem	cncfcn1 23980	Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
Theorem	cncfmptc 23981*	A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
⊢ ((𝐴 ∈ 𝑇 ∧ 𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝐴) ∈ (𝑆–cn→𝑇))
Theorem	cncfmptid 23982*	The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
⊢ ((𝑆 ⊆ 𝑇 ∧ 𝑇 ⊆ ℂ) → (𝑥 ∈ 𝑆 ↦ 𝑥) ∈ (𝑆–cn→𝑇))
Theorem	cncfmpt1f 23983*	Composition of continuous functions. –cn→ analogue of cnmpt11f 22723. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐹‘𝐴)) ∈ (𝑋–cn→ℂ))
Theorem	cncfmpt2f 23984*	Composition of continuous functions. –cn→ analogue of cnmpt12f 22725. (Contributed by Mario Carneiro, 3-Sep-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→ℂ))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→ℂ))
Theorem	cncfmpt2ss 23985*	Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝑋–cn→𝑆))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝑋–cn→𝑆))    &   ⊢ 𝑆 ⊆ ℂ    &   ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋–cn→𝑆))
Theorem	addccncf 23986*	Adding a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 + 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	idcncf 23987	The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 23982 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)    ⇒   ⊢ 𝐹 ∈ (ℂ–cn→ℂ)
Theorem	sub1cncf 23988*	Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐴))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	sub2cncf 23989*	Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐹 = (𝑥 ∈ ℂ ↦ (𝐴 − 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))
Theorem	cdivcncf 23990*	Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))    ⇒   ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ ((ℂ ∖ {0})–cn→ℂ))
Theorem	negcncf 23991*	The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ -𝑥)    ⇒   ⊢ (𝐴 ⊆ ℂ → 𝐹 ∈ (𝐴–cn→ℂ))
Theorem	negfcncf 23992*	The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥))    ⇒   ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ))
Theorem	abscncfALT 23993	Absolute value is continuous. Alternate proof of abscncf 23970. (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)
Theorem	cncfcnvcn 23994	Rewrite cmphaushmeo 22859 for functions on the complex numbers. (Contributed by Mario Carneiro, 19-Feb-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝑋)    ⇒   ⊢ ((𝐾 ∈ Comp ∧ 𝐹 ∈ (𝑋–cn→𝑌)) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑌–cn→𝑋)))
Theorem	expcncf 23995*	The power function on complex numbers, for fixed exponent N, is continuous. Similar to expcn 23941. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ))
Theorem	cnmptre 23996*	Lemma for iirevcn 23999 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ 𝑅 = (TopOpen‘ℂfld)    &   ⊢ 𝐽 = ((topGen‘ran (,)) ↾t 𝐴)    &   ⊢ 𝐾 = ((topGen‘ran (,)) ↾t 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐹) ∈ (𝑅 Cn 𝑅))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐽 Cn 𝐾))
Theorem	cnmpopc 23997*	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 = (topGen‘ran (,))    &   ⊢ 𝑀 = (𝑅 ↾t (𝐴[,]𝐵))    &   ⊢ 𝑁 = (𝑅 ↾t (𝐵[,]𝐶))    &   ⊢ 𝑂 = (𝑅 ↾t (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐶))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐵 ∧ 𝑦 ∈ 𝑋)) → 𝐷 = 𝐸)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵), 𝑦 ∈ 𝑋 ↦ 𝐷) ∈ ((𝑀 ×t 𝐽) Cn 𝐾))    &   ⊢ (𝜑 → (𝑥 ∈ (𝐵[,]𝐶), 𝑦 ∈ 𝑋 ↦ 𝐸) ∈ ((𝑁 ×t 𝐽) Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐶), 𝑦 ∈ 𝑋 ↦ if(𝑥 ≤ 𝐵, 𝐷, 𝐸)) ∈ ((𝑂 ×t 𝐽) Cn 𝐾))
Theorem	iirev 23998	Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,]1) → (1 − 𝑋) ∈ (0[,]1))
Theorem	iirevcn 23999	The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
⊢ (𝑥 ∈ (0[,]1) ↦ (1 − 𝑥)) ∈ (II Cn II)
Theorem	iihalf1 24000	Map the first half of II into II. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑋 ∈ (0[,](1 / 2)) → (2 · 𝑋) ∈ (0[,]1))