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

Theoremnmhmnghm 22601 A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇))

Theoremnmhmghm 22602 A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
(𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremisnmhm2 22603 A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)       ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝑁𝐹) ∈ ℝ))

Theoremnmhmcl 22604 A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)       (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁𝐹) ∈ ℝ)

Theoremidnmhm 22605 The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑉 = (Base‘𝑆)       (𝑆 ∈ NrmMod → ( I ↾ 𝑉) ∈ (𝑆 NMHom 𝑆))

Theorem0nmhm 22606 The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
𝑉 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐹 = (Scalar‘𝑆)    &   𝐺 = (Scalar‘𝑇)       ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇))

Theoremnmhmco 22607 The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
((𝐹 ∈ (𝑇 NMHom 𝑈) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 NMHom 𝑈))

Theoremnmhmplusg 22608 The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
+ = (+g𝑇)       ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹𝑓 + 𝐺) ∈ (𝑆 NMHom 𝑇))

12.4.10  Topology on the reals

Theoremqtopbaslem 22609 The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)
𝑆 ⊆ ℝ*       ((,) “ (𝑆 × 𝑆)) ∈ TopBases

Theoremqtopbas 22610 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
((,) “ (ℚ × ℚ)) ∈ TopBases

Theoremretopbas 22611 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
ran (,) ∈ TopBases

Theoremretop 22612 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
(topGen‘ran (,)) ∈ Top

Theoremuniretop 22613 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
ℝ = (topGen‘ran (,))

Theoremretopon 22614 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGen‘ran (,)) ∈ (TopOn‘ℝ)

Theoremretps 22615 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}       𝐾 ∈ TopSp

Theoremiooretop 22616 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
(𝐴(,)𝐵) ∈ (topGen‘ran (,))

Theoremicccld 22617 Closed intervals are closed sets of the standard topology on . (Contributed by FL, 14-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))

Theoremicopnfcld 22618 Right-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))

Theoremiocmnfcld 22619 Left-unbounded closed intervals are closed sets of the standard topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
(𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))

Theoremqdensere 22620 is dense in the standard topology on . (Contributed by NM, 1-Mar-2007.)
((cls‘(topGen‘ran (,)))‘ℚ) = ℝ

Theoremcnmetdval 22621 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐷 = (abs ∘ − )       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴𝐵)))

Theoremcnmet 22622 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
(abs ∘ − ) ∈ (Met‘ℂ)

Theoremcnxmet 22623 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ − ) ∈ (∞Met‘ℂ)

Theoremcnbl0 22624 Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐷 = (abs ∘ − )       (𝑅 ∈ ℝ* → (abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))

Theoremcnblcld 22625* 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})

Theoremcnfldms 22626 The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ MetSp

Theoremcnfldxms 22627 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ ∞MetSp

Theoremcnfldtps 22628 The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
fld ∈ TopSp

Theoremcnfldnm 22629 The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
abs = (norm‘ℂfld)

Theoremcnngp 22630 The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmGrp

Theoremcnnrg 22631 The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
fld ∈ NrmRing

Theoremcnfldtopn 22632 The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 = (MetOpen‘(abs ∘ − ))

Theoremcnfldtopon 22633 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ (TopOn‘ℂ)

Theoremcnfldtop 22634 The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Top

Theoremcnfldhaus 22635 The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Haus

Theoremunicntop 22636 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)

Theoremcnopn 22637 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)

Theoremzringnrg 22638 The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ NrmRing

Theoremremetdval 22639 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 22640 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 22641 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))       𝐷 ∈ (∞Met‘ℝ)

Theorembl2ioo 22642 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 22643 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 22644 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 22645 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 22646 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 22647 is dense in . (Contributed by NM, 24-Aug-2007.)
𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘𝐷)       ((cls‘𝐽)‘ℚ) = ℝ

Theoremblcvx 22648 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 22649 The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.)
(topGen‘ran (,)) ∈ Haus

Theoremtgqioo 22650 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 22651 The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
(topGen‘ran (,)) ∈ 2nd𝜔

Theoremresubmet 22652 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 22653 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 22654 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 22655 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 22656 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 22657 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 22658 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 22659 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 22660 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 22661 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 22662 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 22663 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (topGen‘ran (,))       ℤ ∈ (Clsd‘𝐽)

Theoremrecld2 22664 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 22665 The integers are a closed set in the topology on . (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐽 = (TopOpen‘ℂfld)       ℤ ∈ (Clsd‘𝐽)

Theoremzdis 22666 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 22667 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 22668* 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 22669 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 22670 The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ Perf

Theoremiccntr 22671 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 22672* Lemma for icccmp 22675. (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 22673* Lemma for icccmp 22675. (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 22674* Lemma for icccmp 22675. (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 22675 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 22676 Lemma for reconn 22678. 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 22677* Lemma for reconn 22678. (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 22678* 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 22679 Corollary of reconn 22678. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
(topGen‘ran (,)) ∈ Conn

Theoremiccconn 22680 A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn)

Theoremopnreen 22681 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 22682 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 22683 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 22684* 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 22685 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 22686 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 22687 The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 22688 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 22688 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 22689 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 22690 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 22691 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 22692* Continuity of the metric function; analogue of cnmpt12f 21517 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 22693* Continuity of the metric function; analogue of cnmpt22f 21526 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 22694 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 22695 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 22696 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 22697* 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 22698* 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 22699* 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 22700* 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)

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