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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cnnrg 24801 | The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| ⊢ ℂfld ∈ NrmRing | ||
| Theorem | cnfldtopn 24802 | The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 = (MetOpen‘(abs ∘ − )) | ||
| Theorem | cnfldtopon 24803 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ (TopOn‘ℂ) | ||
| Theorem | cnfldtop 24804 | The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Top | ||
| Theorem | cnfldhaus 24805 | The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Haus | ||
| Theorem | unicntop 24806 | 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 24807 | 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 | cnn0opn 24808 | The set of nonzero complex numbers is open with respect to the standard topology on complex numbers. (Contributed by SN, 7-Oct-2025.) |
| ⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂfld) | ||
| Theorem | zringnrg 24809 | The ring of integers is a normed ring. (Contributed by AV, 13-Jun-2019.) |
| ⊢ ℤring ∈ NrmRing | ||
| Theorem | remetdval 24810 | 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 24811 | 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 24812 | The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
| ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) ⇒ ⊢ 𝐷 ∈ (∞Met‘ℝ) | ||
| Theorem | bl2ioo 24813 | 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 24814 | 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 24815 | 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 24816 | 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 24817 | 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 24818 | ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.) |
| ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((cls‘𝐽)‘ℚ) = ℝ | ||
| Theorem | blcvx 24819 | 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 24820 | The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.) |
| ⊢ (topGen‘ran (,)) ∈ Haus | ||
| Theorem | tgqioo 24821 | 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 24822 | The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
| ⊢ (topGen‘ran (,)) ∈ 2ndω | ||
| Theorem | resubmet 24823 | 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 24824 | 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 24825 | 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 | tgioo4 24826 | The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | ||
| Theorem | tgioo3 24827 | 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 24828 | 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 24829 | 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 24830 | 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 24831 | 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 24832 | 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 24833 | 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 24834 | 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 24835 | 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 24836 | 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 24837 | 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 24838 | 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 24839 | 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 24840* | 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 24841 | 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 24842 | The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Perf | ||
| Theorem | iccntr 24843 | 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 24844* | Lemma for icccmp 24847. (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 24845* | Lemma for icccmp 24847. (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 24846* | Lemma for icccmp 24847. (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 24847 | 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 24848 | Lemma for reconn 24850. 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 24849* | Lemma for reconn 24850. (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 24850* | 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 24851 | Corollary of reconn 24850. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
| ⊢ (topGen‘ran (,)) ∈ Conn | ||
| Theorem | iccconn 24852 | A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn) | ||
| Theorem | opnreen 24853 | 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 24854 | 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 24855 | 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 24856* | 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 24857 | 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 24858 | 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 24859 | The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 24860 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 24860 | 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 24861 | 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 24862 | 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 24863 | 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 24864* | Continuity of the metric function; analogue of cnmpt12f 23674 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 24865* | Continuity of the metric function; analogue of cnmpt22f 23683 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 24866 | 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 24867 | 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 24868 | 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 24869* | 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 24870* | 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 24871* | 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 24872* | 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) | ||
| Theorem | metdstri 24873* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐴) ≤ ((𝐴𝐷𝐵) +𝑒 (𝐹‘𝐵))) | ||
| Theorem | metdsle 24874* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑋)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵)) | ||
| Theorem | metdsre 24875* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) ⇒ ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝐹:𝑋⟶ℝ) | ||
| Theorem | metdseq0 24876* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆))) | ||
| Theorem | metdscnlem 24877* | Lemma for metdscn 24878. (Contributed by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐶 = (dist‘ℝ*𝑠) & ⊢ 𝐾 = (MetOpen‘𝐶) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ⊆ 𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → (𝐴𝐷𝐵) < 𝑅) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴) +𝑒 -𝑒(𝐹‘𝐵)) < 𝑅) | ||
| Theorem | metdscn 24878* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐶 = (dist‘ℝ*𝑠) & ⊢ 𝐾 = (MetOpen‘𝐶) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | metdscn2 24879* | 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 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | metnrmlem1a 24880* | Lemma for metnrm 24884. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑇) → (0 < (𝐹‘𝐴) ∧ if(1 ≤ (𝐹‘𝐴), 1, (𝐹‘𝐴)) ∈ ℝ+)) | ||
| Theorem | metnrmlem1 24881* | Lemma for metnrm 24884. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) | ||
| Theorem | metnrmlem2 24882* | Lemma for metnrm 24884. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) & ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ⇒ ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) | ||
| Theorem | metnrmlem3 24883* | Lemma for metnrm 24884. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) & ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) & ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑇 ↦ (𝑥𝐷𝑦)), ℝ*, < )) & ⊢ 𝑉 = ∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)) | ||
| Theorem | metnrm 24884 | 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) | ||
| Theorem | metreg 24885 | A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.) |
| ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg) | ||
| Theorem | addcnlem 24886* | Lemma for addcn 24887, subcn 24888, and mulcn 24889. (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‘(𝑢 − 𝑏)) < 𝑦 ∧ (abs‘(𝑣 − 𝑐)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝑏 + 𝑐))) < 𝑎)) ⇒ ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
| Theorem | addcn 24887 | 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) | ||
| Theorem | subcn 24888 | 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) | ||
| Theorem | mulcn 24889 | 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.) Usage of this theorem is discouraged because it depends on ax-mulf 11235. Use mpomulcn 24891 instead. (New usage is discouraged.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
| Theorem | divcnOLD 24890 | Obsolete version of divcn 24892 as of 6-Apr-2025. (Contributed by Mario Carneiro, 12-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0})) ⇒ ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽) | ||
| Theorem | mpomulcn 24891* | Complex number multiplication is a continuous function. Version of mulcn 24889 using maps-to notation, which does not require ax-mulf 11235. (Contributed by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) | ||
| Theorem | divcn 24892 | Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0})) ⇒ ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽) | ||
| Theorem | cnfldtgp 24893 | 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.) |
| ⊢ ℂfld ∈ TopGrp | ||
| Theorem | fsumcn 24894* | 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) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | fsum2cn 24895* | Version of fsumcn 24894 for two-argument mappings. (Contributed by Mario Carneiro, 6-May-2014.) |
| ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ ((𝐽 ×t 𝐿) Cn 𝐾)) | ||
| Theorem | expcn 24896* | 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.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | divccn 24897* | Division by a nonzero constant is a continuous operation. (Contributed by Mario Carneiro, 5-May-2014.) Avoid ax-mulf 11235. (Revised by GG, 16-Mar-2025.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | expcnOLD 24898* | Obsolete version of expcn 24896 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | divccnOLD 24899* | Obsolete version of divccn 24897 as of 6-Apr-2025. (Contributed by Mario Carneiro, 5-May-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℂ ↦ (𝑥 / 𝐴)) ∈ (𝐽 Cn 𝐽)) | ||
| Theorem | sqcn 24900* | 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) | ||
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