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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsszcld 22601 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 22602* 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 22603 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 22604 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Perf

Theoremiccntr 22605 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 22606* Lemma for icccmp 22609. (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 22607* Lemma for icccmp 22609. (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 22608* Lemma for icccmp 22609. (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 22609 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 22610 Lemma for reconn 22612. 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 22611* Lemma for reconn 22612. (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 22612* 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 22613 Corollary of reconn 22612. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
(topGen‘ran (,)) ∈ Conn

Theoremiccconn 22614 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)

Theoremopnreen 22615 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 22616 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 22617 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 22618* 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 22619 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 22620 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 22621 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 22622 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 22622 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 22623 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 22624 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 22625 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 22626* Continuity of the metric function; analogue of cnmpt12f 21450 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 22627* Continuity of the metric function; analogue of cnmpt22f 21459 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 22628 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 22629 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 22630 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 22631* 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 22632* 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 22633* 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 22634* 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 22635* 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 22636* 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 22637* 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 22638* 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 22639* Lemma for metdscn 22640. (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 22640* 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 22641* 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 22642* Lemma for metnrm 22646. (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 22643* Lemma for metnrm 22646. (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 22644* Lemma for metnrm 22646. (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 22645* Lemma for metnrm 22646. (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 22646 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 22647 A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg)

Theoremaddcnlem 22648* Lemma for addcn 22649, subcn 22650, and mulcn 22651. (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 22649 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 22650 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 22651 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 22652 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 22653 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 22654* 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 22655* Version of fsumcn 22654 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 22656* 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 22657* 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 22658* 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 22659 Extend class notation with the unit interval.
class II

Syntaxccncf 22660 Extend class notation to include the operation which returns a class of continuous complex functions.
class cn

Definitiondf-ii 22661 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 22662* 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 22663 The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.)
II ∈ (TopOn‘(0[,]1))

Theoremiitop 22664 The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.)
II ∈ Top

Theoremiiuni 22665 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 22666 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
II = ((topGen‘ran (,)) ↾t (0[,]1))

Theoremdfii3 22667 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 22668 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐼 = (ℂflds (0[,]1))       II = (TopOpen‘𝐼)

Theoremdfii5 22669 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))

Theoremiicmp 22670 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
II ∈ Comp

Theoremiiconn 22671 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
II ∈ Conn

Theoremcncfval 22672* 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 22673* 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 22674* Version of elcncf 22673 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 22675 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Theoremcncfrss2 22676 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Theoremcncff 22677 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐹 ∈ (𝐴cn𝐵) → 𝐹:𝐴𝐵)

Theoremcncfi 22678* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
((𝐹 ∈ (𝐴cn𝐵) ∧ 𝐶𝐴𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑤𝐶)) < 𝑧 → (abs‘((𝐹𝑤) − (𝐹𝐶))) < 𝑅))

Theoremelcncf1di 22679* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+))    &   (𝜑 → (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))       (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵)))

Theoremelcncf1ii 22680* Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.)
𝐹:𝐴𝐵    &   ((𝑥𝐴𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)    &   (((𝑥𝐴𝑤𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥𝑤)) < 𝑍 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴cn𝐵))

Theoremrescncf 22681 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
(𝐶𝐴 → (𝐹 ∈ (𝐴cn𝐵) → (𝐹𝐶) ∈ (𝐶cn𝐵)))

Theoremcncffvrn 22682 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 ∈ (𝐴cn𝐵)) → (𝐹 ∈ (𝐴cn𝐶) ↔ 𝐹:𝐴𝐶))

Theoremcncfss 22683 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
((𝐵𝐶𝐶 ⊆ ℂ) → (𝐴cn𝐵) ⊆ (𝐴cn𝐶))

Theoremclimcncf 22684 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺:𝑍𝐴)    &   (𝜑𝐺𝐷)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐹𝐺) ⇝ (𝐹𝐷))

Theoremabscncf 22685 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
abs ∈ (ℂ–cn→ℝ)

Theoremrecncf 22686 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℜ ∈ (ℂ–cn→ℝ)

Theoremimcncf 22687 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
ℑ ∈ (ℂ–cn→ℝ)

Theoremcjcncf 22688 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
∗ ∈ (ℂ–cn→ℂ)

Theoremmulc1cncf 22689* 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))

Theoremdivccncf 22690* 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))

Theoremcncfco 22691 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.)
(𝜑𝐹 ∈ (𝐴cn𝐵))    &   (𝜑𝐺 ∈ (𝐵cn𝐶))       (𝜑 → (𝐺𝐹) ∈ (𝐴cn𝐶))

Theoremcncfmet 22692 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) → (𝐴cn𝐵) = (𝐽 Cn 𝐾))

Theoremcncfcn 22693 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽t 𝐴)    &   𝐿 = (𝐽t 𝐵)       ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = (𝐾 Cn 𝐿))

Theoremcncfcn1 22694 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)

Theoremcncfmptc 22695* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
((𝐴𝑇𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝐴) ∈ (𝑆cn𝑇))

Theoremcncfmptid 22696* The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))

Theoremcncfmpt1f 22697* Composition of continuous functions. cn analogue of cnmpt11f 21448. (Contributed by Mario Carneiro, 3-Sep-2014.)
(𝜑𝐹 ∈ (ℂ–cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2f 22698* Composition of continuous functions. cn analogue of cnmpt12f 21450. (Contributed by Mario Carneiro, 3-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   (𝜑𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn→ℂ))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn→ℂ))       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn→ℂ))

Theoremcncfmpt2ss 22699* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
𝐽 = (TopOpen‘ℂfld)    &   𝐹 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝑋cn𝑆))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝑋cn𝑆))    &   𝑆 ⊆ ℂ    &   ((𝐴𝑆𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝑆)       (𝜑 → (𝑥𝑋 ↦ (𝐴𝐹𝐵)) ∈ (𝑋cn𝑆))

Theoremaddccncf 22700* 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))

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