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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tgioo 23401 | 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 23402 | ℚ is dense in ℝ. (Contributed by NM, 24-Aug-2007.) |
⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((cls‘𝐽)‘ℚ) = ℝ | ||
Theorem | blcvx 23403 | 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 23404 | The standard topology on the reals is Hausdorff. (Contributed by NM, 8-Mar-2007.) |
⊢ (topGen‘ran (,)) ∈ Haus | ||
Theorem | tgqioo 23405 | 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 23406 | The standard topology on the reals is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (topGen‘ran (,)) ∈ 2ndω | ||
Theorem | resubmet 23407 | 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 23408 | 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 23409 | 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 23410 | 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 23411 | 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 23412 | 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 23413 | 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 23414 | 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 23415 | 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 23416 | 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 23417 | 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 23418 | 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 23419 | 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 23420 | 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 23421 | 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 23422 | 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 23423* | 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 23424 | 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 23425 | The complex numbers are a perfect space. (Contributed by Mario Carneiro, 26-Dec-2016.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ 𝐽 ∈ Perf | ||
Theorem | iccntr 23426 | 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 23427* | Lemma for icccmp 23430. (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 23428* | Lemma for icccmp 23430. (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 23429* | Lemma for icccmp 23430. (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 23430 | 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 23431 | Lemma for reconn 23433. 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 23432* | Lemma for reconn 23433. (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 23433* | 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 23434 | Corollary of reconn 23433. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
⊢ (topGen‘ran (,)) ∈ Conn | ||
Theorem | iccconn 23435 | A closed interval is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾t (𝐴[,]𝐵)) ∈ Conn) | ||
Theorem | opnreen 23436 | 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 23437 | 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 23438 | 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 23439* | 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 23440 | 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 23441 | 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 23442 | The metric function of an extended metric space is always continuous in the topology generated by it. In this variation of xmetdcn 23443 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 23443 | 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 23444 | The metric function of a metric space is always continuous in the topology generated by it. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (topGen‘ran (,)) ⇒ ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) | ||
Theorem | metdcn 23445 | 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 23446 | 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 23447* | Continuity of the metric function; analogue of cnmpt12f 22271 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 23448* | Continuity of the metric function; analogue of cnmpt22f 22280 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 23449 | 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 23450 | 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 23451 | 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 23452* | 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 23453* | 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 23454* | 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 23455* | 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 23456* | 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 23457* | 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 23458* | 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 23459* | 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 23460* | Lemma for metdscn 23461. (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 23461* | 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 23462* | 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 23463* | Lemma for metnrm 23467. (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 23464* | Lemma for metnrm 23467. (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 23465* | Lemma for metnrm 23467. (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 23466* | Lemma for metnrm 23467. (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 23467 | 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 23468 | A metric space is regular. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Reg) | ||
Theorem | addcnlem 23469* | Lemma for addcn 23470, subcn 23471, and mulcn 23472. (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 23470 | 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 23471 | 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 23472 | 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) | ||
Theorem | divcn 23473 | Complex number division is a continuous function, when the second argument is nonzero. (Contributed by Mario Carneiro, 12-Aug-2014.) |
⊢ 𝐽 = (TopOpen‘ℂfld) & ⊢ 𝐾 = (𝐽 ↾t (ℂ ∖ {0})) ⇒ ⊢ / ∈ ((𝐽 ×t 𝐾) Cn 𝐽) | ||
Theorem | cnfldtgp 23474 | 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 23475* | 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 23476* | Version of fsumcn 23475 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 23477* | 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 ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) | ||
Theorem | divccn 23478* | 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)) | ||
Theorem | sqcn 23479* | 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) | ||
Syntax | cii 23480 | Extend class notation with the unit interval. |
class II | ||
Syntax | ccncf 23481 | Extend class notation to include the operation which returns a class of continuous complex functions. |
class –cn→ | ||
Definition | df-ii 23482 | 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)))) | ||
Definition | df-cncf 23483* | 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 ∈ (𝑏 ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+ ∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)}) | ||
Theorem | iitopon 23484 | The unit interval is a topological space. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ II ∈ (TopOn‘(0[,]1)) | ||
Theorem | iitop 23485 | The unit interval is a topological space. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ II ∈ Top | ||
Theorem | iiuni 23486 | The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.) |
⊢ (0[,]1) = ∪ II | ||
Theorem | dfii2 23487 | Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ II = ((topGen‘ran (,)) ↾t (0[,]1)) | ||
Theorem | dfii3 23488 | Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐽 = (TopOpen‘ℂfld) ⇒ ⊢ II = (𝐽 ↾t (0[,]1)) | ||
Theorem | dfii4 23489 | Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐼 = (ℂfld ↾s (0[,]1)) ⇒ ⊢ II = (TopOpen‘𝐼) | ||
Theorem | dfii5 23490 | The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1)))) | ||
Theorem | iicmp 23491 | The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.) |
⊢ II ∈ Comp | ||
Theorem | iiconn 23492 | The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
⊢ II ∈ Conn | ||
Theorem | cncfval 23493* | 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) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)}) | ||
Theorem | elcncf 23494* | 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 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦)))) | ||
Theorem | elcncf2 23495* | Version of elcncf 23494 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.) |
⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴–cn→𝐵) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝑥)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝑥))) < 𝑦)))) | ||
Theorem | cncfrss 23496 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐴 ⊆ ℂ) | ||
Theorem | cncfrss2 23497 | Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.) |
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | ||
Theorem | cncff 23498 | A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐹:𝐴⟶𝐵) | ||
Theorem | cncfi 23499* | Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.) |
⊢ ((𝐹 ∈ (𝐴–cn→𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ ℝ+) → ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑤 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑤) − (𝐹‘𝐶))) < 𝑅)) | ||
Theorem | elcncf1di 23500* | Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 26-Nov-2007.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ℝ+) → 𝑍 ∈ ℝ+)) & ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → ((abs‘(𝑥 − 𝑤)) < 𝑍 → (abs‘((𝐹‘𝑥) − (𝐹‘𝑤))) < 𝑦))) ⇒ ⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → 𝐹 ∈ (𝐴–cn→𝐵))) |
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