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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremioo2bl 23401 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 23402 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 23403 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 23404 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 23405 is dense in . (Contributed by NM, 24-Aug-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘𝐷)       ((cls‘𝐽)‘ℚ) = ℝ
 
Theoremblcvx 23406 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 23407 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
(topGen‘ran (,)) ∈ Haus
 
Theoremtgqioo 23408 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 23409 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(topGen‘ran (,)) ∈ 2ndω
 
Theoremresubmet 23410 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 23411 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 23412 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 23413 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 23414 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 23415 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 23416 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 23417 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 23418 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 23419 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 23420 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 23421 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (topGen‘ran (,))       ℤ ∈ (Clsd‘𝐽)
 
Theoremrecld2 23422 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 23423 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℤ ∈ (Clsd‘𝐽)
 
Theoremzdis 23424 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 23425 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 23426* 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 23427 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 23428 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Perf
 
Theoremiccntr 23429 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 23430* Lemma for icccmp 23433. (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 23431* Lemma for icccmp 23433. (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 23432* Lemma for icccmp 23433. (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 23433 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 23434 Lemma for reconn 23436. 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) ∧ (𝑋𝐴𝑌𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴)
 
Theoremreconnlem2 23435* Lemma for reconn 23436. (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 23436* 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 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥[,]𝑦) ⊆ 𝐴))
 
Theoremretopconn 23437 Corollary of reconn 23436. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
(topGen‘ran (,)) ∈ Conn
 
Theoremiccconn 23438 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)
 
Theoremopnreen 23439 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 23440 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 23441 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 23442* 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 23443 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 23444 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 23445 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 23446 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 23446 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 23447 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 23448 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 23449 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 23450* Continuity of the metric function; analogue of cnmpt12f 22274 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 23451* Continuity of the metric function; analogue of cnmpt22f 22283 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 23452 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))
 
Theoremngnmcncn 23453 The norm of a normed group is a continuous function to . (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))
 
Theoremabscn 23454 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 23455* 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 23456* 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 23457* 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 23458* 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 23459* 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 23460* 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 23461* 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 23462* 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 23463* Lemma for metdscn 23464. (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 23464* 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 23465* 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 23466* Lemma for metnrm 23470. (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 23467* Lemma for metnrm 23470. (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 23468* Lemma for metnrm 23470. (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 23469* Lemma for metnrm 23470. (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) = ∅))
 
Theoremmetnrm 23470 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 23471 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)
 
Theoremaddcnlem 23472* Lemma for addcn 23473, subcn 23474, and mulcn 23475. (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 23473 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 23474 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)
 
Theoremmulcn 23475 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)
 
Theoremdivcn 23476 Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t (ℂ ∖ {0}))        / ∈ ((𝐽 ×t 𝐾) Cn 𝐽)
 
Theoremcnfldtgp 23477 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.)
fld ∈ TopGrp
 
Theoremfsumcn 23478* 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)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → (𝑥𝑋𝐵) ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑥𝑋 ↦ Σ𝑘𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
 
Theoremfsum2cn 23479* Version of fsumcn 23478 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.)
𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   ((𝜑𝑘𝐴) → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ Σ𝑘𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾))
 
Theoremexpcn 23480* 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 ↦ (𝑥𝑁)) ∈ (𝐽 Cn 𝐽))
 
Theoremdivccn 23481* 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))
 
Theoremsqcn 23482* 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
 
Syntaxcii 23483 Extend class notation with the unit interval.
class II
 
Syntaxccncf 23484 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn
 
Definitiondf-ii 23485 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))))
 
Definitiondf-cncf 23486* 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 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦𝑎 ((abs‘(𝑥𝑦)) < 𝑑 → (abs‘((𝑓𝑥) − (𝑓𝑦))) < 𝑒)})
 
Theoremiitopon 23487 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
II ∈ (TopOn‘(0[,]1))
 
Theoremiitop 23488 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
II ∈ Top
 
Theoremiiuni 23489 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
(0[,]1) = II
 
Theoremdfii2 23490 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
II = ((topGen‘ran (,)) ↾t (0[,]1))
 
Theoremdfii3 23491 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       II = (𝐽t (0[,]1))
 
Theoremdfii4 23492 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐼 = (ℂflds (0[,]1))       II = (TopOpen‘𝐼)
 
Theoremdfii5 23493 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
 
Theoremiicmp 23494 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp
 
Theoremiiconn 23495 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn
 
Theoremcncfval 23496* 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) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
 
Theoremelcncf 23497* 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 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
 
Theoremelcncf2 23498* Version of elcncf 23497 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝑥)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝑥))) < 𝑦))))
 
Theoremcncfrss 23499 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
 
Theoremcncfrss2 23500 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45259
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