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Theorem List for Metamath Proof Explorer - 22301-22400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnfldnm 22301 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
abs = (norm‘ℂfld)
 
Theoremcnngp 22302 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmGrp
 
Theoremcnnrg 22303 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmRing
 
Theoremcnfldtopn 22304 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 = (MetOpen‘(abs ∘ − ))
 
Theoremcnfldtopon 22305 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ (TopOn‘ℂ)
 
Theoremcnfldtop 22306 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Top
 
Theoremcnfldhaus 22307 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Haus
 
Theoremzringnrg 22308 The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ NrmRing
 
Theoremremetdval 22309 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‘(𝐴𝐵)))
 
Theoremremet 22310 The absolute value metric determines a metric space on the reals. (Contributed by NM, 10-Feb-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (Met‘ℝ)
 
Theoremrexmet 22311 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (∞Met‘ℝ)
 
Theorembl2ioo 22312 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) = ((𝐴𝐵)(,)(𝐴 + 𝐵)))
 
Theoremioo2bl 22313 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)((𝐵𝐴) / 2)))
 
Theoremioo2blex 22314 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))
 
Theoremblssioo 22315 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 (,)
 
Theoremtgioo 22316 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
 
Theoremqdensere2 22317 is dense in . (Contributed by NM, 24-Aug-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘𝐷)       ((cls‘𝐽)‘ℚ) = ℝ
 
Theoremblcvx 22318 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*) ∧ (𝐴𝑆𝐵𝑆𝑇 ∈ (0[,]1))) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ 𝑆)
 
Theoremrehaus 22319 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
(topGen‘ran (,)) ∈ Haus
 
Theoremtgqioo 22320 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
 
Theoremre2ndc 22321 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(topGen‘ran (,)) ∈ 2nd𝜔
 
Theoremresubmet 22322 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 = (𝑅t 𝐴))
 
Theoremtgioo2 22323 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 (,)) = (𝐽t ℝ)
 
Theoremrerest 22324 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 → (𝐽t 𝐴) = (𝑅t 𝐴))
 
Theoremtgioo3 22325 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
 
Theoremxrtgioo 22326 The topology on the extended reals coincides with the standard topology on the reals, when restricted to . (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐽 = ((ordTop‘ ≤ ) ↾t ℝ)       (topGen‘ran (,)) = 𝐽
 
Theoremxrrest 22327 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝑋 = (ordTop‘ ≤ )    &   𝑅 = (topGen‘ran (,))       (𝐴 ⊆ ℝ → (𝑋t 𝐴) = (𝑅t 𝐴))
 
Theoremxrrest2 22328 The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝑋 = (ordTop‘ ≤ )       (𝐴 ⊆ ℝ → (𝐽t 𝐴) = (𝑋t 𝐴))
 
Theoremxrsxmet 22329 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*)
 
Theoremxrsdsre 22330 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))
 
Theoremxrsblre 22331 Any ball of the metric of the extended reals centered on an element of is entirely contained in . (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐷 = (dist‘ℝ*𝑠)       ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ)
 
Theoremxrsmopn 22332 The metric on the extended reals generates a topology, but this does not match the order topology on *; 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
 
Theoremzcld 22333 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (topGen‘ran (,))       ℤ ∈ (Clsd‘𝐽)
 
Theoremrecld2 22334 The real numbers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℝ ∈ (Clsd‘𝐽)
 
Theoremzcld2 22335 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℤ ∈ (Clsd‘𝐽)
 
Theoremzdis 22336 The integers are a discrete set in the topology on . (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝐽t ℤ) = 𝒫 ℤ
 
Theoremsszcld 22337 Every subset of the integers are closed in the topology on . (Contributed by Mario Carneiro, 6-Jul-2017.)
𝐽 = (TopOpen‘ℂfld)       (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽))
 
Theoremreperflem 22338* 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)    &   ((𝑢𝑆𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ 𝑆)    &   𝑆 ⊆ ℂ       (𝐽t 𝑆) ∈ Perf
 
Theoremreperf 22339 The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       (𝐽t ℝ) ∈ Perf
 
Theoremcnperf 22340 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Perf
 
Theoremiccntr 22341 The interior of a closed interval in the standard topology on is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
 
Theoremicccmplem1 22342* Lemma for icccmp 22345. (Contributed by Mario Carneiro, 18-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)       (𝜑 → (𝐴𝑆 ∧ ∀𝑦𝑆 𝑦𝐵))
 
Theoremicccmplem2 22343* Lemma for icccmp 22345. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)    &   (𝜑𝑉𝑈)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐺(ball‘𝐷)𝐶) ⊆ 𝑉)    &   𝐺 = sup(𝑆, ℝ, < )    &   𝑅 = if((𝐺 + (𝐶 / 2)) ≤ 𝐵, (𝐺 + (𝐶 / 2)), 𝐵)       (𝜑𝐵𝑆)
 
Theoremicccmplem3 22344* Lemma for icccmp 22345. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))    &   𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ 𝑧}    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝐴[,]𝐵) ⊆ 𝑈)       (𝜑𝐵𝑆)
 
Theoremicccmp 22345 A closed interval in is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
𝐽 = (topGen‘ran (,))    &   𝑇 = (𝐽t (𝐴[,]𝐵))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
 
Theoremreconnlem1 22346 Lemma for reconn 22348. 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) ∈ Con) ∧ (𝑋𝐴𝑌𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴)
 
Theoremreconnlem2 22347* Lemma for reconn 22348. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.)
(𝜑𝐴 ⊆ ℝ)    &   (𝜑𝑈 ∈ (topGen‘ran (,)))    &   (𝜑𝑉 ∈ (topGen‘ran (,)))    &   (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥[,]𝑦) ⊆ 𝐴)    &   (𝜑𝐵 ∈ (𝑈𝐴))    &   (𝜑𝐶 ∈ (𝑉𝐴))    &   (𝜑 → (𝑈𝑉) ⊆ (ℝ ∖ 𝐴))    &   (𝜑𝐵𝐶)    &   𝑆 = sup((𝑈 ∩ (𝐵[,]𝐶)), ℝ, < )       (𝜑 → ¬ 𝐴 ⊆ (𝑈𝑉))
 
Theoremreconn 22348* 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) ∈ Con ↔ ∀𝑥𝐴𝑦𝐴 (𝑥[,]𝑦) ⊆ 𝐴))
 
Theoremretopcon 22349 Corollary of reconn 22348. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
(topGen‘ran (,)) ∈ Con
 
Theoremiccconn 22350 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Con)
 
Theoremopnreen 22351 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)
 
Theoremrectbntr0 22352 A countable subset of the reals has empty interior. (Contributed by Mario Carneiro, 26-Jul-2014.)
((𝐴 ⊆ ℝ ∧ 𝐴 ≼ ℕ) → ((int‘(topGen‘ran (,)))‘𝐴) = ∅)
 
Theoremxrge0gsumle 22353 A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝐵 ∈ (𝒫 𝐴 ∩ Fin))    &   (𝜑𝐶𝐵)       (𝜑 → (𝐺 Σg (𝐹𝐶)) ≤ (𝐺 Σg (𝐹𝐵)))
 
Theoremxrge0tsms 22354* 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*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑠))), ℝ*, < )       (𝜑 → (𝐺 tsums 𝐹) = {𝑆})
 
Theoremxrge0tsms2 22355 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 * or or [0, +∞). It is true for 0 ∪ {+∞}, however, or more generally any additive submonoid of [0, +∞) with +∞ adjoined. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝐺 = (ℝ*𝑠s (0[,]+∞))       ((𝐴𝑉𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1𝑜)
 
Theoremmetdcnlem 22356 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)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑌𝑋)    &   (𝜑𝑍𝑋)    &   (𝜑 → (𝐴𝐷𝑌) < (𝑅 / 2))    &   (𝜑 → (𝐵𝐷𝑍) < (𝑅 / 2))       (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) < 𝑅)
 
Theoremxmetdcn2 22357 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 22358 we use the metric topology instead of the order topology on *, 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))
 
Theoremxmetdcn 22358 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))
 
Theoremmetdcn2 22359 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))
 
Theoremmetdcn 22360 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))
 
Theoremmsdcn 22361 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))
 
Theoremcnmpt1ds 22362* Continuity of the metric function; analogue of cnmpt12f 21182 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 (,))    &   (𝜑𝐺 ∈ MetSp)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐷𝐵)) ∈ (𝐾 Cn 𝑅))
 
Theoremcnmpt2ds 22363* Continuity of the metric function; analogue of cnmpt22f 21191 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 (,))    &   (𝜑𝐺 ∈ MetSp)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴𝐷𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅))
 
Theoremnmcn 22364 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))
 
Theoremabscn 22365 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)
 
Theoremmetdsval 22366* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (𝐴𝑋 → (𝐹𝐴) = inf(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
 
Theoremmetdsf 22367* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → 𝐹:𝑋⟶(0[,]+∞))
 
Theoremmetdsge 22368* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))
 
Theoremmetds0 22369* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑆) → (𝐹𝐴) = 0)
 
Theoremmetdstri 22370* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹𝐵)))
 
Theoremmetdsle 22371* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑆𝐵𝑋)) → (𝐹𝐵) ≤ (𝐴𝐷𝐵))
 
Theoremmetdsre 22372* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹:𝑋⟶ℝ)
 
Theoremmetdseq0 22373* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑋) → ((𝐹𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆)))
 
Theoremmetdscnlem 22374* Lemma for metdscn 22375. (Contributed by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝐴𝐷𝐵) < 𝑅)       (𝜑 → ((𝐹𝐴) +𝑒 -𝑒(𝐹𝐵)) < 𝑅)
 
Theoremmetdscn 22375* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetdscn2 22376* 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 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐾 = (TopOpen‘ℂfld)       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetnrmlem1a 22377* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑𝐴𝑇) → (0 < (𝐹𝐴) ∧ if(1 ≤ (𝐹𝐴), 1, (𝐹𝐴)) ∈ ℝ+))
 
Theoremmetnrmlem1 22378* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑 ∧ (𝐴𝑆𝐵𝑇)) → if(1 ≤ (𝐹𝐵), 1, (𝐹𝐵)) ≤ (𝐴𝐷𝐵))
 
Theoremmetnrmlem2 22379* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))       (𝜑 → (𝑈𝐽𝑇𝑈))
 
Theoremmetnrmlem3 22380* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)
𝐹 = (𝑥𝑋 ↦ inf(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))    &   𝐺 = (𝑥𝑋 ↦ inf(ran (𝑦𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝑉 = 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2))       (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
 
TheoremmetdsvalOLD 22381* Value of the "distance to a set" function. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 4-Sep-2015.) Obsolete version of metdsval 22366 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (𝐴𝑋 → (𝐹𝐴) = sup(ran (𝑦𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < ))
 
TheoremmetdsfOLD 22382* 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.) Obsolete version of metdsf 22367 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → 𝐹:𝑋⟶(0[,]+∞))
 
TheoremmetdsgeOLD 22383* 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.) Obsolete version of metdsge 22368 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅))
 
Theoremmetds0OLD 22384* 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.) Obsolete version of metds0 22369 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑆) → (𝐹𝐴) = 0)
 
TheoremmetdstriOLD 22385* 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.) Obsolete version of metdstri 22370 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹𝐵)))
 
TheoremmetdsleOLD 22386* 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.) Obsolete version of metdsle 22371 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝐴𝑆𝐵𝑋)) → (𝐹𝐵) ≤ (𝐴𝐷𝐵))
 
TheoremmetdsreOLD 22387* 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.) Obsolete version of metdsre 22372 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹:𝑋⟶ℝ)
 
Theoremmetdseq0OLD 22388* 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.) Obsolete version of metdseq0 22373 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋𝐴𝑋) → ((𝐹𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆)))
 
TheoremmetdscnlemOLD 22389* Lemma for metdscn 22375. (Contributed by Mario Carneiro, 4-Sep-2015.) Obsolete version of metdscnlem 22374 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑋)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → (𝐴𝐷𝐵) < 𝑅)       (𝜑 → ((𝐹𝐴) +𝑒 -𝑒(𝐹𝐵)) < 𝑅)
 
TheoremmetdscnOLD 22390* 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.) Obsolete version of metdscn 22375 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐶 = (dist‘ℝ*𝑠)    &   𝐾 = (MetOpen‘𝐶)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetdscn2OLD 22391* 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.) Obsolete version of metdscn2 22376 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   𝐾 = (TopOpen‘ℂfld)       ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremmetnrmlem1aOLD 22392* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) Obsolete version of metnrmlem1a 22377 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑𝐴𝑇) → (0 < (𝐹𝐴) ∧ if(1 ≤ (𝐹𝐴), 1, (𝐹𝐴)) ∈ ℝ+))
 
Theoremmetnrmlem1OLD 22393* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) Obsolete version of metnrmlem1 22378 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)       ((𝜑 ∧ (𝐴𝑆𝐵𝑇)) → if(1 ≤ (𝐹𝐵), 1, (𝐹𝐵)) ≤ (𝐴𝐷𝐵))
 
Theoremmetnrmlem2OLD 22394* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) Obsolete version of metnrmlem2 22379 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))       (𝜑 → (𝑈𝐽𝑇𝑈))
 
Theoremmetnrmlem3OLD 22395* Lemma for metnrm 22396. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) Obsolete version of metnrmlem3 22380 as of 30-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐹 = (𝑥𝑋 ↦ sup(ran (𝑦𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆 ∈ (Clsd‘𝐽))    &   (𝜑𝑇 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝑆𝑇) = ∅)    &   𝑈 = 𝑡𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹𝑡), 1, (𝐹𝑡)) / 2))    &   𝐺 = (𝑥𝑋 ↦ sup(ran (𝑦𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < ))    &   𝑉 = 𝑠𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺𝑠), 1, (𝐺𝑠)) / 2))       (𝜑 → ∃𝑧𝐽𝑤𝐽 (𝑆𝑧𝑇𝑤 ∧ (𝑧𝑤) = ∅))
 
Theoremmetnrm 22396 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)
 
Theoremmetreg 22397 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
 
Theoremaddcnlem 22398* Lemma for addcn 22399, subcn 22400, and mulcn 22401. (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‘(𝑢𝑏)) < 𝑦 ∧ (abs‘(𝑣𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremaddcn 22399 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)
 
Theoremsubcn 22400 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)
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