HomeTheorem List (p. 227 of 429)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | nmhmnghm 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)) | ||
| Theorem | nmhmghm 22602 | A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
| ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
| Theorem | isnmhm2 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) ↔ (𝑁‘𝐹) ∈ ℝ)) | ||
| Theorem | nmhmcl 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) → (𝑁‘𝐹) ∈ ℝ) | ||
| Theorem | idnmhm 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)) | ||
| Theorem | 0nmhm 22606 | The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
| ⊢ 𝑉 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐹 = (Scalar‘𝑆) & ⊢ 𝐺 = (Scalar‘𝑇) ⇒ ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇)) | ||
| Theorem | nmhmco 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)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NMHom 𝑈)) | ||
| Theorem | nmhmplusg 22608 | The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.) |
| ⊢ + = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘𝑓 + 𝐺) ∈ (𝑆 NMHom 𝑇)) | ||
| Theorem | qtopbaslem 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 | ||
| Theorem | qtopbas 22610 | The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.) |
| ⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases | ||
| Theorem | retopbas 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 | ||
| Theorem | retop 22612 | The standard topology on the reals. (Contributed by FL, 4-Jun-2007.) |
| ⊢ (topGen‘ran (,)) ∈ Top | ||
| Theorem | uniretop 22613 | The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.) |
| ⊢ ℝ = ∪ (topGen‘ran (,)) | ||
| Theorem | retopon 22614 | The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) | ||
| Theorem | retps 22615 | The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.) |
| ⊢ 𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩} ⇒ ⊢ 𝐾 ∈ TopSp | ||
| Theorem | iooretop 22616 | Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) |
| ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | ||
| Theorem | icccld 22617 | Closed intervals are closed sets of the standard topology on ℝ. (Contributed by FL, 14-Sep-2007.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,)))) | ||
| Theorem | icopnfcld 22618 | Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | ||
| Theorem | iocmnfcld 22619 | Left-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) | ||
| Theorem | qdensere 22620 | ℚ is dense in the standard topology on ℝ. (Contributed by NM, 1-Mar-2007.) |
| ⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ | ||
| Theorem | cnmetdval 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‘(𝐴 − 𝐵))) | ||
| Theorem | cnmet 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) | ||
| Theorem | cnxmet 22623 | The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ) | ||
| Theorem | cnbl0 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)) | ||
| Theorem | cnblcld 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}) | ||
| Theorem | cnfldms 22626 | The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ ℂfld ∈ MetSp | ||
| Theorem | cnfldxms 22627 | The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ ℂfld ∈ ∞MetSp | ||
| Theorem | cnfldtps 22628 | The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ ℂfld ∈ TopSp | ||
| Theorem | cnfldnm 22629 | The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ abs = (norm‘ℂfld) | ||
| Theorem | cnngp 22630 | The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ ℂfld ∈ NrmGrp | ||
| Theorem | cnnrg 22631 | The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ ℂfld ∈ NrmRing | ||
| Theorem | cnfldtopn 22632 | The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) | ||
| Theorem | cnfldtopon 22633 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ (TopOn‘ℂ) | ||
| Theorem | cnfldtop 22634 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Top | ||
| Theorem | cnfldhaus 22635 | The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Haus | ||
| Theorem | unicntop 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) | ||
| Theorem | cnopn 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) | ||
| Theorem | zringnrg 22638 | The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.) |
| ⊢ ℤring ∈ NrmRing | ||
| Theorem | remetdval 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‘(𝐴 − 𝐵))) | ||
| Theorem | remet 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) | ||
| Theorem | rexmet 22641 | The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐷 ∈ (∞Met‘ℝ) | ||
| Theorem | bl2ioo 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) = ((𝐴 − 𝐵)(,)(𝐴 + 𝐵))) | ||
| Theorem | ioo2bl 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)((𝐵 − 𝐴) / 2))) | ||
| Theorem | ioo2blex 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)) | ||
| Theorem | blssioo 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 (,) | ||
| Theorem | tgioo 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 | ||
| Theorem | qdensere2 22647 | ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.) |
| ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((cls‘𝐽)‘ℚ) = ℝ | ||
| Theorem | blcvx 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*) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝑇 ∈ (0[,]1))) → ((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵)) ∈ 𝑆) | ||
| Theorem | rehaus 22649 | The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.) |
| ⊢ (topGen‘ran (,)) ∈ Haus | ||
| Theorem | tgqioo 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 | ||
| Theorem | re2ndc 22651 | The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ (topGen‘ran (,)) ∈ 2nd𝜔 | ||
| Theorem | resubmet 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 = (𝑅 ↾t 𝐴)) | ||
| Theorem | tgioo2 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 (,)) = (𝐽 ↾t ℝ) | ||
| Theorem | rerest 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 → (𝐽 ↾t 𝐴) = (𝑅 ↾t 𝐴)) | ||
| Theorem | tgioo3 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 | ||
| Theorem | xrtgioo 22656 | The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Mario Carneiro, 3-Sep-2015.) |
| ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t ℝ) ⇒ ⊢ (topGen‘ran (,)) = 𝐽 | ||
| Theorem | xrrest 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 → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) | ||
| Theorem | xrrest2 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 → (𝐽 ↾t 𝐴) = (𝑋 ↾t 𝐴)) | ||
| Theorem | xrsxmet 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*) | ||
| Theorem | xrsdsre 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)) | ||
| Theorem | xrsblre 22661 | Any ball of the metric of the extended reals centered on an element of ℝ is entirely contained in ℝ. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐷 = (dist‘ℝ*𝑠) ⇒ ⊢ ((𝑃 ∈ ℝ ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ ℝ) | ||
| Theorem | xrsmopn 22662 | The metric on the extended reals generates a topology, but this does not match the order topology on ℝ*; for example {+∞} is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐷 = (dist‘ℝ*𝑠) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (ordTop‘ ≤ ) ⊆ 𝐽 | ||
| Theorem | zcld 22663 | The integers are a closed set in the topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ 𝐽 = (topGen‘ran (,)) ⇒ ⊢ ℤ ∈ (Clsd‘𝐽) | ||
| Theorem | recld2 22664 | The real numbers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ ℝ ∈ (Clsd‘𝐽) | ||
| Theorem | zcld2 22665 | The integers are a closed set in the topology on ℂ. (Contributed by Mario Carneiro, 17-Feb-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ ℤ ∈ (Clsd‘𝐽) | ||
| Theorem | zdis 22666 | The integers are a discrete set in the topology on ℂ. (Contributed by Mario Carneiro, 19-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐽 ↾t ℤ) = 𝒫 ℤ | ||
| Theorem | sszcld 22667 | Every subset of the integers are closed in the topology on ℂ. (Contributed by Mario Carneiro, 6-Jul-2017.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐴 ⊆ ℤ → 𝐴 ∈ (Clsd‘𝐽)) | ||
| Theorem | reperflem 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) & ⊢ ((𝑢 ∈ 𝑆 ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ 𝑆) & ⊢ 𝑆 ⊆ ℂ ⇒ ⊢ (𝐽 ↾t 𝑆) ∈ Perf | ||
| Theorem | reperf 22669 | The real numbers are a perfect subset of the complex numbers. (Contributed by Mario Carneiro, 26-Dec-2016.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐽 ↾t ℝ) ∈ Perf | ||
| Theorem | cnperf 22670 | The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Perf | ||
| Theorem | iccntr 22671 | The interior of a closed interval in the standard topology on ℝ is the corresponding open interval. (Contributed by Mario Carneiro, 1-Sep-2014.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) | ||
| Theorem | icccmplem1 22672* | Lemma for icccmp 22675. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵)) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧} & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) ⇒ ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝐵)) | ||
| Theorem | icccmplem2 22673* | Lemma for icccmp 22675. (Contributed by Mario Carneiro, 13-Jun-2014.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵)) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧} & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → (𝐺(ball‘𝐷)𝐶) ⊆ 𝑉) & ⊢ 𝐺 = sup(𝑆, ℝ, < ) & ⊢ 𝑅 = if((𝐺 + (𝐶 / 2)) ≤ 𝐵, (𝐺 + (𝐶 / 2)), 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑆) | ||
| Theorem | icccmplem3 22674* | Lemma for icccmp 22675. (Contributed by Mario Carneiro, 13-Jun-2014.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵)) & ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ ∃𝑧 ∈ (𝒫 𝑈 ∩ Fin)(𝐴[,]𝑥) ⊆ ∪ 𝑧} & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑆) | ||
| Theorem | icccmp 22675 | A closed interval in ℝ is compact. (Contributed by Mario Carneiro, 13-Jun-2014.) |
| ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝑇 = (𝐽 ↾t (𝐴[,]𝐵)) ⇒ ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp) | ||
| Theorem | reconnlem1 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) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋[,]𝑌) ⊆ 𝐴) | ||
| Theorem | reconnlem2 22677* | Lemma for reconn 22678. (Contributed by Jeff Hankins, 17-Aug-2009.) (Proof shortened by Mario Carneiro, 9-Sep-2015.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝑈 ∈ (topGen‘ran (,))) & ⊢ (𝜑 → 𝑉 ∈ (topGen‘ran (,))) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∩ 𝐴)) & ⊢ (𝜑 → 𝐶 ∈ (𝑉 ∩ 𝐴)) & ⊢ (𝜑 → (𝑈 ∩ 𝑉) ⊆ (ℝ ∖ 𝐴)) & ⊢ (𝜑 → 𝐵 ≤ 𝐶) & ⊢ 𝑆 = sup((𝑈 ∩ (𝐵[,]𝐶)), ℝ, < ) ⇒ ⊢ (𝜑 → ¬ 𝐴 ⊆ (𝑈 ∪ 𝑉)) | ||
| Theorem | reconn 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 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥[,]𝑦) ⊆ 𝐴)) | ||
| Theorem | retopconn 22679 | Corollary of reconn 22678. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
| ⊢ (topGen‘ran (,)) ∈ Conn | ||
| Theorem | iccconn 22680 | A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn) | ||
| Theorem | opnreen 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) | ||
| Theorem | rectbntr0 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) = ∅) | ||
| Theorem | xrge0gsumle 22683 | A finite sum in the nonnegative extended reals is monotonic in the support. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∩ Fin)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ≤ (𝐺 Σg (𝐹 ↾ 𝐵))) | ||
| Theorem | xrge0tsms 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*𝑠 ↾s (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) & ⊢ 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))), ℝ*, < ) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝑆}) | ||
| Theorem | xrge0tsms2 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 ℝ* or ℝ or [0, +∞). It is true for ℕ0 ∪ {+∞}, however, or more generally any additive submonoid of [0, +∞) with +∞ adjoined. (Contributed by Mario Carneiro, 13-Sep-2015.) |
| ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1𝑜) | ||
| Theorem | metdcnlem 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) & ⊢ (𝜑 → 𝑍 ∈ 𝑋) & ⊢ (𝜑 → (𝐴𝐷𝑌) < (𝑅 / 2)) & ⊢ (𝜑 → (𝐵𝐷𝑍) < (𝑅 / 2)) ⇒ ⊢ (𝜑 → ((𝐴𝐷𝐵)𝐶(𝑌𝐷𝑍)) < 𝑅) | ||
| Theorem | xmetdcn2 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 ℝ*, which makes the theorem a bit stronger. Since +∞ is an isolated point in the metric topology, this is saying that for any points 𝐴, 𝐵 which are an infinite distance apart, there is a product neighborhood around ⟨𝐴, 𝐵⟩ such that 𝑑(𝑎, 𝑏) = +∞ for any 𝑎 near 𝐴 and 𝑏 near 𝐵, i.e. the distance function is locally constant +∞. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐶 = (dist‘ℝ*𝑠) & ⊢ 𝐾 = (MetOpen‘𝐶) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) | ||
| Theorem | xmetdcn 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)) | ||
| Theorem | metdcn2 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)) | ||
| Theorem | metdcn 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)) | ||
| Theorem | msdcn 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)) | ||
| Theorem | cnmpt1ds 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 (,)) & ⊢ (𝜑 → 𝐺 ∈ MetSp) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴𝐷𝐵)) ∈ (𝐾 Cn 𝑅)) | ||
| Theorem | cnmpt2ds 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 (,)) & ⊢ (𝜑 → 𝐺 ∈ MetSp) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴𝐷𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝑅)) | ||
| Theorem | nmcn 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)) | ||
| Theorem | ngnmcncn 22695 | The norm of a normed group is a continuous function to ℂ. (Contributed by NM, 12-Aug-2007.) (Revised by AV, 17-Oct-2021.) |
| ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝐺 ∈ NrmGrp → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | abscn 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) | ||
| Theorem | metdsval 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝐹‘𝐴) = inf(ran (𝑦 ∈ 𝑆 ↦ (𝐴𝐷𝑦)), ℝ*, < )) | ||
| Theorem | metdsf 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) | ||
| Theorem | metdsge 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑅 ∈ ℝ*) → (𝑅 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑅)) = ∅)) | ||
| Theorem | metds0 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0) | ||
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