P. 232 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnblcld 23101*	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 23102	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ MetSp
Theorem	cnfldxms 23103	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ ∞MetSp
Theorem	cnfldtps 23104	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ TopSp
Theorem	cnfldnm 23105	The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ abs = (norm‘ℂfld)
Theorem	cnngp 23106	The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmGrp
Theorem	cnnrg 23107	The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmRing
Theorem	cnfldtopn 23108	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
Theorem	cnfldtopon 23109	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cnfldtop 23110	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	cnfldhaus 23111	The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Haus
Theorem	unicntop 23112	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	cnopn 23113	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	zringnrg 23114	The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ NrmRing
Theorem	remetdval 23115	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 23116	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 23117	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ)
Theorem	bl2ioo 23118	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 23119	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 23120	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 23121	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 23122	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	qdensere2 23123	ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.)
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((cls‘𝐽)‘ℚ) = ℝ
Theorem	blcvx 23124	An open ball in the complex numbers is a convex set. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
⊢ 𝑆 = (𝑃(ball‘(abs ∘ − ))𝑅)    ⇒   ⊢ (((𝑃 ∈ ℂ ∧ 𝑅 ∈ ℝ*) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ 𝑆)
Theorem	rehaus 23125	The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
⊢ (topGen‘ran (,)) ∈ Haus
Theorem	tgqioo 23126	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	re2ndc 23127	The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (topGen‘ran (,)) ∈ 2nd𝜔
Theorem	resubmet 23128	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	tgioo2 23129	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 23130	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	tgioo3 23131	The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Thierry Arnoux, 3-Jul-2019.)
⊢ 𝐽 = (TopOpen‘ℝfld)    ⇒   ⊢ (topGen‘ran (,)) = 𝐽
Theorem	xrtgioo 23132	The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t ℝ)    ⇒   ⊢ (topGen‘ran (,)) = 𝐽
Theorem	xrrest 23133	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = (ordTop‘ ≤ )    &   ⊢ 𝑅 = (topGen‘ran (,))    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴))
Theorem	xrrest2 23134	The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝑋 = (ordTop‘ ≤ )    ⇒   ⊢ (𝐴 ⊆ ℝ → (𝐽 ↾t 𝐴) = (𝑋 ↾t 𝐴))
Theorem	xrsxmet 23135	The metric on the extended reals is a proper extended metric. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ 𝐷 ∈ (∞Met‘ℝ*)
Theorem	xrsdsre 23136	The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
Theorem	xrsblre 23137	Any ball of the metric of the extended reals centered on an element of ℝ is entirely contained in ℝ. (Contributed by Mario Carneiro, 4-Sep-2015.)
⊢ 𝐷 = (dist‘ℝ*𝑠)    ⇒   ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)
Theorem	xrsmopn 23138	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 23139	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 23140	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 23141	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 23142	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 23143	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 23144*	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 23145	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 23146	The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Perf
Theorem	iccntr 23147	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 23148*	Lemma for icccmp 23151. (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 23149*	Lemma for icccmp 23151. (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 23150*	Lemma for icccmp 23151. (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 23151	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 23152	Lemma for reconn 23154. 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 23153*	Lemma for reconn 23154. (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 23154*	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 23155	Corollary of reconn 23154. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ (topGen‘ran (,)) ∈ Conn
Theorem	iccconn 23156	A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
Theorem	opnreen 23157	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 23158	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 23159	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 23160*	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 23161	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 23162	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 23163	The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 23164 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 23164	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 23165	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 23166	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 23167	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 23168*	Continuity of the metric function; analogue of cnmpt12f 21993 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 23169*	Continuity of the metric function; analogue of cnmpt22f 22002 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 23170	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 23171	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 23172	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 23173*	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 23174*	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 23175*	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 23176*	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 23177*	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 23178*	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 23179*	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 23180*	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 23181*	Lemma for metdscn 23182. (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 23182*	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 23183*	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 23184*	Lemma for metnrm 23188. (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 23185*	Lemma for metnrm 23188. (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 23186*	Lemma for metnrm 23188. (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 23187*	Lemma for metnrm 23188. (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 23188	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 23189	A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
Theorem	addcnlem 23190*	Lemma for addcn 23191, subcn 23192, and mulcn 23193. (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 23191	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 23192	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 23193	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 23194	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 23195	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 23196*	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 23197*	Version of fsumcn 23196 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 23198*	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 23199*	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 23200*	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)